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Number Theory

Number Theory
Proceedings of the Turku Symposium
on Number Theory in Memory of Kustaa Inkeri
May 31-June 4,1999
Editors
Matti Jutila and Tauno Metsänkylä
W
DE
G_
Walter de Gruyter · Berlin · New York 2001

Editors
Tauno Metsänkylä Matti Jutila
Department of Mathematics Department of Mathematics
University of Turku University of Turku
20014 Turku 20014 Turku
Finland Finland
Mathematics Subject Classification 2000:
11-06
© Printed on acid-free paper which falls within the guidelines of the
ANSI to ensure permanence and durability.
Library of Congress — Cataloging-in-Publication Data
Turku Symposium on Number Theory (1999 : Turku, Finland)
Number theory : proceedings of the Turku Symposium on
Number Theory in memory of Kustaa Inkeri, May 31 - June
4, 1999 / editors, Matti Jutila and Tauno Metsänkylä.
p. cm.
ISBN 3 11 016481 7 (alk. paper)
1. Number theory - Congresses. I. Jutila, M. (Matti)
II. Metsänkylä, Tauno. III. Title.
QA241 .T85 1999
512'.7-dc21 00-065759
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Number theory : proceedings of the Turku Symposium on
Number Theory in Memory of Kustaa Inkeri, May 31 — June 4,
1999 / ed. Matti Jutila and Tauno Metsänkylä. — Berlin ;
New York : de Gruyter, 2001
ISBN 3-11-016481-7
© Copyright 2001 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy, recording or any information storage and retrieval system, without permis-
sion in writing from the publisher.
Printed in Germany.
Typeset using the authors' TEX files: I. Zimmermann, Freiburg.
Printing: WB-Druck GmbH & Co., Rieden/Allgäu. Binding: Lüderitz & Bauer-GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.

Preface
The conference Turku Symposium on Number Theory in Memory of Kustaa Inkeri
was held in Turku, Finland, from May 31 to June 4, 1999, and the present volume
contains selected contributions of the participants. All in all, there were 71 par-
ticipants representing 21 countries. This meeting, the main event of the Number
Theory Year 1999 organized by the Rolf Nevanlinna Institute, was dedicated to the
memory of Kustaa Inkeri (1908-1997) who as a Professor of Mathematics at the
University of Turku laid the foundation of the number theoretic research tradition in
Finland. Plenary lectures were given by E. Bach (Madison), P. Bundschuh (Köln),
G. Frey (Essen), M. Ν. Huxley (Cardiff), Y. Motohashi (Tokyo), W. Narkiewicz
(Wroclaw), A. van der Poorten (Sydney), and P. Ribenboim (Kingston); the plenary
lecture by A. Ivic (Beograd), who due to unfortunate circumstances could not attend
the symposium, was presented by M. Jutila. In addition, there were about 40 con-
tributed talks. The leisure program included a visit to the Sibelius Museum with a
chamber concert, a boat trip to Naantali with a sightseeing tour of the summer res-
idence Kultaranta of the President of Finland, and finally a banquet in a pavilion
situated on the little island Pikku-Pukki in the Turku archipelago.
The subject of the symposium was number theory in a broad sense covering ele-
mentary, algebraic, analytic, computational, and applied number theory, Diophantine
equations and approximation, and arithmetic algebraic geometry. The following top-
ics, among others, are discussed in the papers included in this volume: Lucas and
Pell numbers, equations of the Catalan and Fermât type, elliptic curves, the ABC-
conjecture, Abelian fields, the arithmetic nature of certain specific numbers or func-
tion values, exponential sums, sieve methods, the Riemann zeta-function and other
Dirichlet series, and the spectral theory of automorphic functions with its arithmetical
applications. All papers were refereed. We sincerely thank the authors and referees
for their contributions.
The organizers gratefully acknowledge support from the following sponsors: the
Rolf Nevanlinna Institute, the University of Turku, the Turku University Foundation,
the Turku Centre of Computer Science (TUCS), and the Nokia Company. Also we wish
to thank all the staff members of the Department of Mathematics and the mathematics
students involved in the arrangements for their efforts for the success of the symposium.
Turku, June 2000 Matti Jutila
Tauno Metsänkylä

Table of Contents
Preface ν
Tauno Metsänkylä
On the life and work of Kustaa Inkeri 1
MasaakiAmou, Masanori Katsurada and Keijo Väänänen
On the values of certain q -hypergeometric series 5
Yann Bugeaud
On the Diophantine equation f^· = yq 19
Peter Bundschuh
Arithmetical properties of the solutions of certain functional equations 25
Karl Dilcher and Kirk Haller
Multiple zeta sums via box splines 43
Gerhard Frey
Galois representations attached to elliptic curves and diophantine problems 71
George Greaves and Martin Huxley
One-sided sifting density hypotheses in Selberg's sieve 105
Peniti Haukkanen
On characterizations of completely multiplicative arithmetical functions 115
Charles Helou
A note on the power residue symbol 125
Martin Huxley and Grigori Kolesnik
Exponential sums with a large second derivative 131
Aleksandar Ivic
Some mean value results for the Riemann zeta-function 145 Terence Jackson Direct proofs of some of Euler's results 163
Matti Jutila
The fourth moment of central values of Hecke series 167

viii Table of Contents
Antanas Laurincikas
The universality of Dirichlet series attached to finite Abelian groups 179
Tapani Matala-aho
An irrationality criterion for p-adic binomial series 193
Kohji Matsumoto
The mean values and the universality of Rankin-Selberg L-functions 201
Tom Meurman
On the binary additive divisor problem 223
Maurice Mignotte
Catalan's equation just before 2000 247
Yoichi Motohashi
New analytic problems over imaginary quadratic number fields 255
Alfred J. van der Poorten
On number theory and Kustaa Inkeri 281
Paulo Ribenboim and Wayne L. McDaniel
On Lucas sequence terms of the form kx2 293
Jonathan W. Sands
Stark's question and Popescu's conjecture for abelian L-functions 305
Indulis Strazdins
Partial Pell numbers 317
List of participants 323

On the life and work of Kustaa Inkeri
Tauno Metsänkylä
Kustaa Inkeri (1908-1997) was Professor of Mathematics at the University of Turku
from 1950 to 1972. He was a pioneer in his research field, number theory, in Finland;
before him there had been only sporadic articles on this area by Finnish mathemati-
cians. Inkeri had seven doctoral students, all of whom have later worked as professors
in different Finnish universities, in most cases as number theorists as well. So there is
every reason to regard Inkeri as the founder of the number theory school in Finland.
Inkeri was born to a farmer's family nearby Turku. He completed his school in
the city and enrolled in the University in 1930. At that time the university, called
the Finnish University of Turku, was a small private institution, while the only state
university in Finland was in Helsinki. (Curiously, there was another small university,
Âbo Akademi, in Turku, having Swedish as its language.) Inkeri's teacher was Kalle
Väisälä, an algebraist but also attracted by number theory. Inkeri finished his doctoral
thesis "Untersuchungen über die Fermatsche Vermutung" [2] in 1946. Meanwhile he
had had periods of heavy military service in the wars of 1939-1940 and 1941-1944.
Inkeri succeeded Väisälä in the chair. Until the end of the fifties the Mathematics
Department remained very modest in size, but it began to grow as the great post-war
age classes were entering the universities. By the time of Inkeri's retiring there were
four full professors and five associate professors (applied mathematics included) and,
of course, a lot of other teachers and staff. However, the burdensome post of the Head
of the Department had been held by Inkeri through all the years.
All his work, whether research, teaching, running the Department or in other ways
contributing to the scientific community, Inkeri took very conscientiously. This meant
that he could not have any special free-time activities. Even during the holidays he
was often occupied by many duties which forced him to put research aside. After
achieving the status of Emeritus he thus felt particularly happy in finally having time
to do what he liked best. More than a third of all his research articles are from this
period.
A more detailed account of Inkeri's person and life can be read in my article in-
cluded in the volume [1] which comprises Inkeri's collected papers. As for a compre-
hensive description and evaluation of these papers, it is a challenging task still awaiting
its author. Here I will just touch a few aspects in Inkeri's extensive production.
Throughout his whole scientific activity Inkeri was keenly interested in the Fermât
problem. He is one of the most quoted authors in Paulo Ribenboim's two monographs

2 Tauno Metsänkylä
on this subject, the "classic" [6] and the recent [7]. One result, frequently cited in the
literature, asserts that in a putative solutions, y, ζ of the Fermât equation χp+yp = zp,
with ρ a prime, one has
min(j:,y) > pip~4, max(;t, y) > \pip~l
thus the solution would be so huge that it would be impossible to ever reach it by
computation. (Of course we now know, thanks to Wiles, that no solution really exists.)
For another striking result, obtained in a joint work with Alfred van der Poorten,
see Poorten's article in the book at hand.
In his very first work [3] after the dissertation Inkeri dealt with a hot question,
the Euclidean quadratic fields Q(s/m). It had been known for some time that there
exist 22 such fields, with — 11 < m < 97, and it was believed that there are no more.
In common efforts of many people the range of dubious values of m was gradually
shrinking to nought. In this process Inkeri took the next to last step by excluding all
such m between 97 and 5000. The last step, settling the range from 5000 to 214, then
followed quickly.
One related theme was that of the Minkowski constant for binary quadratic forms.
Several mathematicians worked on this topic at that time. Inkeri published a series
of articles around it in 1950-1957, the last jointly with Veikko Ennola, his student.
In one of the papers he managed, for example, to improve upon some estimates by
Davenport. That kind of work also led him to some further problems in this area of
Diophantine approximations.
Of the three last articles, appearing in the nineties, two are concerned with Catalan's
equation xp—yq = 1. The former [5] provides an argument very similar to that proving
Fermat's conjecture for regular prime exponents in the "first case". The result says that
if the prime q does not divide the class number of the pth cyclotomic field, then the
solvability of Catalan's equation implies that pq~l = 1 (mod q2). By interchanging
the roles of χ and y one arrives at the condition
pq~l = 1 (mod q2), qp~x = 1 (mod p2),
which looks rather strange and indeed turns out to be extremely rare. One wonders
that nobody had previously come up with the same reasoning! (But, of course, many
brilliant ideas seem simple in retrospect.) In fact Inkeri had been fairly close to it as
early as 1964 [4], Subsequently there has been quite an exciting development in this
question, as reported in the article by Maurice Mignotte in the present volume.
References
[1] Collected Papers of Kustaa Inkeri (ed. by T. Metsänkylä, P. Ribenboim), Queen's Papers
in Pure and Appi. Math. 91, Kingston, Ontario, Canada, 1992.

On the life and work of Kustaa Inkeri 3
[2] Inkeri, K., Untersuchungen über die Fermatsche Vermutung, Ann. Acad. Sei. Fenn. Ser.
A133 (1946), 60 pp.
[3] Inkeri, K„ Über den Euklidischen Algorithmus in quadratischen Zahlkörpern, Ann. Acad.
Sci. Fenn. Ser. A141 (1947), 35 pp.
[4] Inkeri, K„ On Catalan's problem, Acta Arith. 9 ( 1964), 285-290.
[5] Inkeri, K„ On Catalan's conjecture, J. Number Theory 34 (1990), 142-152.
[6] Ribenboim, P., 13 Lectures on Fermat's Last Theorem, Springer-Verlag, 1979.
[7] Ribenboim, P., Fermat's Last Theorem for Amateurs, Springer-Verlag, 1999.
Address of the author:
Department of Mathematics
University of Turku
20014 Turku, Finland
E-mail: [email protected]

On the values of certain q -hypergeometric series
Masaaki Amou* Masanori Katsurada**, and Keijo Vaananen
Abstract. The present paper considers applications to the authors' previous work on the arith-
metical properties of the values of certain functional equations of Poincaré type. We shall give
explicit conditions implying that, for any nonzero a belonging to an algebraic number field Κ
of finite degree over Q, the values φ (a) of a rather general class of q-hypergeometric series φ
do not belong to Κ. The results also contain irrationality measures for these values.
1991 Mathematics Subject Classification: 11J82,11J72,11J61
Dedicated to the Memory ofKustaa Inkeri
1. Introduction
In the previous paper [1] the authors consider arithmetical properties of the values of
functions satisfying the functional equation of Poincaré type
zsf(z) = P(z)f(qz) + Q(z), (1.1)
where s is a positive integer, q is a nonzero element of an algebraic number field Κ,
and P, Q e K[z\. The result extends and quantifies the main result of Duverney [3].
We also applied it to certain <7-hypergeometric series. In particular, we partly extend
the main result of Stihl [5] and quantify a special case of the main result of Bézivin [2].
The purpose of the present paper is to give more precise results on the values of certain
<7-hypergeometric series.
We here recall our previous result. Let A' be an algebraic number field of degree
d over Q and M the set of all places of K. We denote by OK the ring of integers in
K. For any w e M, we normalize the absolute value | I«, of Κ so that \p\w — p~l
for a finite place w lying over the prime number ρ and that \x\w = (x e Q) for
*The author was supported in part by Grant-in-Aid for Scientific Research (No. 11640009), Ministry
of Education, Science, Sports and Culture in Japan.
**The author was supported in part by Grant-in-Aid for Scientific Research (No. 11640038) Ministry of
Education, Science, Sports and Culture in Japan.

6 Masaaki Amou, Masanori Katsurada and Keijo Väänänen
an infinite place w, where | | denotes the ordinary absolute value. We denote by Kw
the completion of Κ with respect to w, and put dw = [Kw : Q«,]. Then, by using the
symbol || || w = | we define the absolute height h (a) of a e Κ by the formula
h{a) = J~[max(l, ||a||uj)·
w
Let ν & M and q € Κ be such that \q\v > 1. We introduce the quantity λ defined by
log h(q)
λ = λ(υ, q) =
log Ili II»'
Note that λ > 1, and λ = 1 if and only if \q\w < 1 for all w φ v. Further, for any
η, 1 < η < 2, and ρ > 0, we define
Α(η, ρ) = p{ 1 + ps/2), Φ(τ?, ρ) = ρ(η - 1)
with
Ρ — 2 — η + ρ, η =
6(2 - η)
Then we proved the following result in [1],
Theorem. LetK, v, and q be as above. Let s be a positive integer, and let Ρ (ζ) e K[z
be of degree t < s such that P(0) φ 0 and
P(q~k) φ 0 for all \t > 0.
Let φ(ζ) = φ(ζ~, s, Ρ) be the function defined by
00 ~s(k)
w-ZpwA'.PiiW· (12)
which is entire on Kv. Assume that there exist η, 1 < η < 2, and ρ > 0 for which
Γ := Φ — (λ — 1)Λ — ηλ>0. (1.3)
(Note that the condition holds when λ = 1.) Then, for nonzero a e Κ, φ (a) (e Kv)
belongs to Κ if and only if a belongs to 8q(s, P), where Sq(s, P) is the set consisting
of all elements a € Κ for which the functional equation
P(z)f(qz)=azsf(z) + P(z) (1.4)
has a solution f(z) in K[z\.
Furthermore, for any nonzero a € Κ not belonging to Sq (s, P) and for any e > 0,
there exists a positive constant c depending only on φ, a, and e such that for all θ e Κ
Iφ(α) - θ\υ > Η-μ~€, Η = max(A(0), c), (1.5)

On the values of certain g-hypergeometric series 7
with
μ -
d(A + Φ)
dv Γ '
(1.6)
We shall say that φ(α) has an irrationality measure μ in K, if (1.5) holds. To give
an idea of μ, we note that the choice η = 3/2, ρ — 6 leads in the case λ = 1 to
_ ¿(122 + 169s)
μ~ \5dv "
In view of the theorem, there remains a problem to determine the set Sq (s, Ρ) for
making the statement of the result more precise. Concerning this, we observed in [1]
the following. If t < s, then 8q(s, Ρ) = {0}. In this case our theorem partly extends
the main result of Stihl [5]. If t = s, then
Sq(s, P)\{0} C {asqk \k > 1},
where as is the coefficient of the highest term of P. We note that in the special case
under consideration the result of Bézivin [2] holds if a i {asqk I k e Ζ}. If Ρ is of
the form P(z) = Azs + Β with nonzero elements A, Β of K, then we know exactly
that
eq(s, P)\{0} = {Aqsk \k> 1}.
In the present paper we aim to investigate this question more generally. Let L be
any number field, and let 8q(s, P) denote the set consisting of all a e L such that the
functional equation (1.4) with q e L and
s
Ρ (ζ) = Σ>ζ'· Φ aszs + αο, ai € L, asao φ 0, (1.7)
ι=0
has a polynomial solution f(z) € L[z]. We shall find that in many cases Sq(s, P) =
{0}. In particular, we pay a special attention to the polynomial
P(z) = (z-qi+Vi)...(z~q1+v°), (1.8)
where vi,..., are non-negative integers. In this case the function (1.2) is a q-
hypergeometric series represented by
00 zn
where (*; q)o = 1 and
{x-,q)n = (\-x)(\-xq)...(\-xqn-i)
for n e Ν. Under these notations, we now state our main results.

8 Masaaki Amou, Masanori Katsurada and Keijo Väänänen
Theorem 1. Let s >2. Let L = Κ bean algebraic number field of finite degree over
Q, and let α,·(χ) e i = 0, 1,..., s, be such that
Let Ρ be a polynomial (1.7), where a¡ = Then Sq(s, P) = {0} for all but
finitely many q e Θ κ orq~l e Θ χ.
This result is rather general, but ineffective. The next result ensures in certain
cases that Sq(s, P) = {0}.
Theorem 2. Let s >2. Then Sq (s, P) = {0} if one of the following conditions holds:
(i )L C M and the polynomial Ρ is oftheform (1.7) satisfying as _ι φ 0, as-2¡ > 0,
a.j_2i-i < 0, andq > 1.
(ii) j=2,LcR and a\ φ 0, α2«ο > 0, q > 1.
Naturally we can apply Theorems 1 and 2 to the special polynomial (1.8). For this
polynomial we can prove furthermore
Theorem 3. Let Ρ be the polynomial (1.8), where s > 2. Then Sq(s, P) = {0} in the
following further cases:
(i) s = 2, L c M and 0 < q < 1.
(ii) s is a prime number, L = Q, either all ι»,· are even or all v, are odd, and q
is a negative integer φ —I or the reciprocal of such a number, excluding the cases
±1±2·ν/-Γ, ±2±ν=7, ±4±v/=7, (±l±v°7)/2, (±3±V^7)/2, ±b-J^O, where
b is a positive integer and D is a square free positive integer such that D = 3 (mod 4)
and 1 + b2D = V for some j e Ν.
By combining our previous theorem stated before and Theorems 1, 2, or 3, we
have the following
Theorem 4. Let s > 2. Let Κ be an algebraic number field of degree d over Q,
ν a place of Κ, and q an element of Κ such that \q\v > 1 and the condition (1.3)
is satisfied with some η and p. Let φ{ζ) be the function (1.2) with the polynomial
P(z) of type (1.7) or (1.8). Suppose that q and P(z) satisfy some of the conditions
of Theorems 2 or 3 (where L = K) implying 8q(s, P) = {0}. Then, for any nonzero
a e Κ, φ(α) (e Kv) does not belong to Κ and has an irrationality measure (1.6)
in K. The same is true for all but finitely many q if the conditions of Theorem 1 are
valid.
a^ldK-ií-Dífl^í-l) - 2αΐ(-1)αΐ_2(-Ό) φ 0. (1.9)

On the values of certain g-hypergeometric series 9
It is interesting to note that there exist certain cases for which 8q(s, Ρ) φ {0}.
Indeed, we have
β, (2, (z-q)2)D
{O,?3} in L = Q(^) withq = —3,
{0,?4} in L = Q(^) withq =
-1±V=7
Then the function (1.2) on C in each case has a zero at —27 or (—1 ± 3-y/—7)/2,
respectively.
To prove our theorems, it is necessary to know when the functional equation (1.4)
may have a polynomial solution. In the next section we give preliminary results for
this purpose. Then, in Section 3, we prove our theorems.
2. Preliminary results
We shall denote here by L any number field. Let s be a positive integer greater than
one, and let Ρ be a polynomial (1.7). Let q and a be nonzero elements of L. Then
we consider the functional equation (1.4), which has the unique solution f(z) in the
formal power series ring L[[z]]. We wish to find a condition on q and a such that
f(z) e LVzl
Let us assume that (1.4) has a polynomial solution f(z) of degree n. It is easy to
see that η > 1. By comparing the highest coefficients of both sides of (1.4), we have
a = asqn. (2.1)
We can rewrite (1.4) as
P(z){f(qz)-\} = azsf(z),
which implies that P(z) divides f(z) and η > s. Hence we have, say,
/ω = P(Z)R(Z) (R(z) e L[z]). (2.2)
We set
m
*ω = ]Γ^<· (m = n-s). (2.3)
i=0
It is easy to see that m > 1 under the assumption P(z) Φ aszs + oq. By inserting
(2.2) into (1.4) and dividing both sides by P(z), we obtain
P(qz)R(qz)=azsR(z) + 1.
Hence, by inserting (1.7), (2.1), and (2.3) into this functional equation and comparing
the coefficients of the corresponding terms of the both sides, we have a system of η

10 Masaaki Amou, Masanori Katsurada and Keijo Väänänen
linear equations for b¡ (i = 0,1,..., m) such as
αφο = 1
αφο + ai-φι-\ \-di-mbm= 0 (i = 1,..., s - 1)
α,¿o + ai-Φι Η l·ai-mbm = asqn~'b¡-s (i — s,... ,n - 1)
with the convention a¡ = 0 for i < 0, i > s. Note that the last equations can be
written as
- qn~l)bis +as-i-s+i + • • · -I-ai-mbm =0 (i = s,..., η - 1).
We denote by An the η χ (m +1) matrix consisting of the coefficients of the system of
linear equations above and by Bn the nx(m+2) matrix which is An with'(l, 0,..., 0)
as the last column. Note that A„ has rank m + 1 = η + 1 — s. Since the functional
equation (1.4) with (1.7) has a polynomial solution /(z) of degree η > s if and only
if the system of linear equations above has a solution b¡ e L (/ = 0,1,..., m), by
the theory of linear equations, we have the following
Criterion. Let « e Ν with η > s. Then the functional equation (1.4) with (1.7) has a
polynomial solution f(z) of degree η € Ν if and only if a satisfies (2.1) and the rank
of Βn is equal to η + 1 — s.
To apply the criterion effectively, let us consider a submatrix Cn, say, of Bn
consisting of the i-th rows of Bn with i = 1, s, s + 1,..., η. We note that Cn is a
square matrix of size η + 2 — s, and that Cn = Bn when s = 2. It follows from
the criterion that if the functional equation (1.4) with (2.1) has a polynomial solution
for some positive integer η > s, then det Cn = 0. In the particular case s = 2,
the functional equation (1.4) with (2.1) has a polynomial solution for some positive
integer η > s if and only if det Bn — 0.
We now set η = s + k with non-negative integer k. By expanding det Cs+k
according to the last column, the determinant becomes
(-1)*+3
«i-l as-2 ••• ao
as(\-qk) as-i ... αϊ oq
as(l-q2) aj-1 as-2
as(\-q) as-
Then, by expanding the determinant above according to the first column, and so on,
we reach the following recursion formula for det Cs+k (k = 0,1,2,... ).
Lemma 1. For any non-negative integer k, we have
k+l 1-2
det Cs+k = 4~la*-i Π(1 - · det Cs+k-i (2.4)
/=! 7=0

On the values of certain g-hypergeometric series 11
with the convention a¡ = 0 for i < 0 and det Cs-\ = 1.
In the special case s = 2, we have
det Bk+i = -a\ det Bk+\ - 02^0(1 - qk) det Bk, (2.5)
with the convention det Bq = 0, det B\ = 1.
The next result gives certain congruence relations for det Cs+k (k = 0,1,2,... ).
Lemma 2. In the polynomial ring Z[ao, • • • ,as,q], we have for any non-negative
integer k
det Cs+k = (-fl,_ i)*+1 (mod 1 - q)
and
_ (as-i~ 2asas-2)^ (mod 1 + q), if k is odd,
aet c s+k = 1 7 k
—«.s-iia^j — 2aiaJ_2)5 (mod 1 + q), ifk is even.
Proof Since q = 1 (mod 1 — q), we have by (2.4)
det Cs+k = (-l)k+3a^¡ = (-fl,_!)k+1 (mod 1 - q),
which is the first congruence relation. We next prove the second congruence relation.
By (2.4) together with q = —\ (mod 1 + q), we have for I > 0
det Cs+u s -fli-i det Cs+u-\ (mod 1 + q)
and
detCs+2£+i = —as-\át\C2s+i-lasas-2àeXCs+2Î-\ (mod 1 + q)
= (a¡-1 - 2asas-2) det Cs+2t-\ (mod 1 + q).
It follows from these two formulas that for I > 0
detCs+2e = —as-\{a^_i - 2asas-2)e (mod 1 + q)
and
det Cs+u+\ = (a2s_x - 2αΐαί_2)ί+1 (mod 1 +q),
which imply the second congruence relation. This completes the proof. •
Assume now that L = Κ, an algebraic number field of finite degree over Q, and
a¡ = a¡(q) e OkW\, i = 0, 1,..., j. By denoting
Yi-ai( 1), S¡ = α,(—1)
for i = 0,1,..., s, we obtain from Lemma 2 the following

12 Masaaki Amou, Masanori Katsurada and Keijo Väänänen
Lemma 3. In the polynomial ring 0¡we have for any non-negative integer k
det Cs+k = (-ft-i)*+1 (mod 1 - q)
and
(<£_j - 2<$Α-2)^ (mod 1 + q), ifk is odd,
det Cs+k =
-Sí-ií^j - 2á,s<Si5-2)J (mod 1 + q), ifk is even.
By using the above notations, we define two subsets of Θ κ by
•Si (Κ) = {β e Θ κ I β\ for some k e Ν}
and
h(K) = {β€θκ\ - 2SsSs-2)k for some k e Ν}.
As a consequence of Lemma 3 we then have the following result, which will be used
for proving Theorems 1 and 3.
Proposition. Let q e Οχ or q~l G Θχ· Assume that the functional equation (1.4)
with (1.7) and (2.1) has a polynomial solution for some positive integer η > s. Then
1 - q € (K) and 1 +q e when q e Θκ, and 1 - q~l € ¿(K) and
1 + q~l e S2(K) when q~l e Θκ·
Proof. By the assumption, we have det C„ = 0. Hence the assertion obviously holds
by Lemma 3 when q e Θκ· We next consider the case q ~1 e Οχ. Let us set η = s+k.
By Lemma 3, there exist polynomials U (x), V(x) e Θ κ M such that
(1 - q)U{q) = Ysk+I
and
(1 +q)V(q) =
- 25,5,-2) ^, if k is odd,
o k
á,-i - lhshs-2)1·, if k is even.
Let u and ν be the degrees of U and V, respectively. Then we can rewrite these
relations as
-(1 -q-X)U{q-l)=q-^Y^l
1Λ _ J q'(v+i\S2s_x -2áA_2)^, if* is odd,
?-(υ+1)5ί_ι(5ί2_1 -23,5,-2)2, if k is even,
where U and V are the reciprocal polynomials of U and V, respectively. This implies
the desired assertion. •

On the values of certain q -hypergeometric series
3. Proof of the theorems
13
In this section we prove Theorems 1, 2, and 3.
ProofofTheorem 1. We denote by £> the set consisting of all elements q e 0jc\{O, ±1}
for which det Cn = 0 for some η > s, and by <£)' the set consisting of all elements
q € K\{0, ±1} with q~l e Θκ for which det Cn = 0 for some η > s. Our task is
to show the sets D and £>' are finite sets. We first show that £> is a finite set. Let us
define a map σ from <© into Κ by
π(ηΛ-\ i1" íVd+í). tih(l-q)<Hl+q),
aKq)~ J (l+q)/(l-q), ifh(l+q)<Hl-q).
Then we claim that there exists a finite set of places S of Κ such that \a(q)\w = 1 for
all q e ¿D, w φ S, and
Γ"[ min(l, ||1 +a(q)\w) < —y-. (3.2)
Ls h{a{q))v
By Corollary 1.2 in Lang [4], Chap. 7, this implies the finiteness of the set σ (SD), from
which the finiteness of the set <0 follows, since a(q') = o(q) if and only if q' = ±q.
By noting (1 — q) + (1 + q) = 2, we have
l+a(q) = J
2/iX+q), if A(1 — q) < h(l + q),
2/(1 -q), ifh(i+q)<Hl-q).
Let q e S) and assume h(l—q) < h(l+q). Leti; e Moo suchthat |l+<?|uj < |1+ί|υ
for all w € Moo, where Moo is the set of all infinite places of Κ. Note that ν depends
on q. Then we have
IU+<T(<7)L < p. (3.3)
Indeed, since 1 + q e Θκ, we have
Hl+q)= Π max(l,||l+ç|U)= ]~[ max(l, ||1 + 9||w).
w&M
Thus
h(i + q)< Π tt + q\îw/d = \i + q\v.
weMx
Since h(a(q)) < h( 1 - q)h(\ + q) < h( 1 + q)2, we obtain h(o(q))V2 < |1 + q\v,
which implies that

14 Masaaki Amou, Masanori Katsurada and Keijo Väänänen
as desired. Similarly, we can show the inequality (3.3) also in the case where h(\+q) <
h(\ — q). Therefore (3.2) holds when S D Moo· Under the assumption (1.9) in the
theorem, it follows from the proposition together with (3.1) that there exists a finite
set 5 of places of Κ for which | a(q) |w = 1 whenever w φ S. By including all infinite
places in S, our claim holds.
To prove the finiteness of the set <£>', we can apply the above argument by replacing
q with q~l. This completes the proof of Theorem 1. •
Proof of Theorem 2. (i) We shall show that the assumptions imply det Cs+k > 0 for
all non-negative integers k. First we note that det Cf_i = 1, det Cs = —as-\ > 0.
By applying induction and the recursion formula (2.4), we then obtain the desired
inequality for all k.
(ii) According as αϊ <0 or a\ > 0, we can easily deduce from the recursion
formula (2.5) with α2αο > 0 that det B¡ > 0 or det B2¡-\ > 0, det Bu < 0 for all
positive integers i, respectively. This shows the assertion, and completes the proof of
Theorem 2. •
Proof of Theorem 3. (i) Under the assumption 0 < q < 1, we show det Bn > 0 for
all η > 2. To this end we prove that
det Bn > qy/qxl,+V2 det Bn-1 (3.4)
for all η > 2. Since (qV] + qV2)/2 > y/qvwe have
det B2 = q(qn + qn) > 2qjq^,
which shows (3.4) for η = 2. Let us assume (3.4) for η with η > 2 and show (3.4)
replacing η with η + 1. By the recursion formula (2.5) and the induction hypothesis,
we have
detfl„+1 = q{qvi+qv*)d*BH-q2+Vi+v>(l-qn-i)detBn-l
> t «-, - g2+v^ · det B"
qjqï7+¡ñ
= qjq^+n det Bn,
as desired.
(ii) Note that the coefficients a¡, i = 0,1,..., s, of the polynomial Ρ (ζ) of the
type (1.8) can be written as a¡ = a¡(q), where α,·(χ) e Z[x]. In particular, we have
s
as(x) = 1, fli-iOO = -χ Σ**' = χ2 Σ *Vi+Vi•
1=1 1 <i<j<s
Then, under the assumption that either all of v, are even or all of v,· are odd, we have
Ys-1 = -s, δ5-ι - ±s, - 25^5^2 = s, (3.5)

On the values of certain q -hypergeometric series 15
where y, = a¡( 1) and <5,· = a¡(—1) as before.
We first consider the case where q is a negative integer with q φ — 1. Suppose that
det Cn — 0 for some η > s. Then we show that (s, q) = (2, —3) or (5, q) = (3, —2).
It follows from the proposition that there exist non-negative integers i and j such that
1 -q=s\ 1 +q = -sj. (3.6)
Hence we have 2 = sl - sK It is easy to see that this holds only when (s, i, j) =
(2,2,1), (3,1,0). The former corresponds to (s, q) = (2, —3) and the latter corre-
sponds to (s, q) = (3, —2).
For the case where q~l is a negative integer with q φ —1, we have (3.6) by
replacing q with q~l. Therefore (s, q~l) must be (2, -3) or (3, —2).
(iii) Let us assume that q is a nonzero imaginary quadratic integer such that
det B„ = 0 for some η >2. We first consider the case where q is of the form
q=a + b*/--D,
where a and b φ 0 are integers and D is a square free positive integer such that
—D = 2 or 3 (mod 4), or equivalently,
£> = 1 or 2 (mod 4).
Since we have (3.5) with s = 2, by the proposition, there exist non-negative integers
i and j such that
\l-q\2 = (l-a)2 + b2D = 2i, |1 + q\2 = (1 + a)2 + b2D = V. (3.7)
If a = 0, then these two equalities coincide. That is,
\+b1D = 2i. (3.8)
We see that i > 0 and that (b, D) = (±1,1) are the solutions of (3.8) with i = 1,
which correspond to q = We show that (3.8) does not hold when i > 2.
Indeed, if (3.8) holds, then b and D must satisfy b = ± 1 (mod 4) and D = 1 (mod 4).
This implies that 1 + b2D = 2 (mod 4), which is inconsistent with (3.8) when i > 2.
Let now a > 0 and set k := j - i > 1. It follows from (3.7) that
(1 + a)2 + b2D = 2λ{(1 — a)2 + b2D}. (3.9)
If a > 6, then, by noting that (1 + a)2 < 2(1 - a)2 and k > 1, (3.9) does not holds.
Hence we may assume that a < 5. Since
(l+a)2 + b2D = (l-a)2 + b2D (mod 2* - 1)
by (3.9), we obtain 4a = 0 (mod 2k — 1), namely,
a = 0 (mod 2* — 1). (3.10)
If k > 2, then we must have k = 2 and a — 3. Hence we deduce 3b2D - 0 from
(3.9), which is impossible. If k = 1, then we have (1 — a)2 -I- b2D = 4a by (3.9).
Hence a = 2l~2 by (3.7), which is possible when a = 1, 2, or 4. If a = 2 or 4, then

16 Masaaki Amou, Masanori Katsurada and Keijo Väänänen
we have b2D = 7 by (3.9), which contradicts the assumption on D. If a = 1, then
we have b2D = 4 by (3.9). Thus we obtain (b, D) = (±2,1), which correspond to
q = 1 ± 2y/-l.
For the case a < 0, we can apply the same argument as above by replacing a with
—a and obtain q = — 1 ± 2λ/—ϊ·
We next consider the case where q is of the form
a + b^D
q = 2 '
where a and b φ 0 are integers with a = b (mod 2) and D is a square free positive
integer such that — D = 1 (mod 4), or equivalently,
D = 3 (mod 4).
By the same reason as in the previous case, there exist non-negative integers i and j
such that
Hence we have
(2 — a)2 + b2D = 2'+2, (2 + a)2 + b2D = 2;+2. (3.11)
If a = 0, then b must be an even number, say 2b'. In this case these two equalities
coincide, that is, 1 + b'2D = 2'. Thus q is of the form q = b'-J—D, where b' is a
nonzero integer.
Let now a > 0 and set k := j — i > 1. It follows from (3.11) that
(2 + a)2 + b2D = 2*{(2 - a)2 + b2D}. (3.12)
If a > 12, then, by noting that (2 + a)2 < 2(2 - a)2 and k > 1, (3.12) does not holds.
If a > 6 and k > 2, then, since (2 + a)2 < 4(2 - a)2, (3.12) does not holds. Hence
we may assume that k = 1 and a < 11, or k > 2 and a < 5. Since
(2 + a)2 + b2D = (2 - a)2 + b2D (mod 2* - 1)
by (3.12), (3.10) holds. If k > 2, then we must have k = 2 and a = 3. Hence
we have b2D = 7 by (3.12), and obtain (b, D) = (±1,7), which correspond to
q = (3 ± y=7)/2. If k = 1, then we have (a - 2)2 + b2D = 8a by (3.12). Hence
α = 2'~3 by (3.10), which is possible when a = 1, 2,4 or 8. If a = 1, then we
have b2D = 7 by (3.12). Thus we obtain (è, D) = (±1,7), which correspond to
q = (1 ± v/z7)/2. If α = 2, then we have b2D = 16 by (3.12), which contradicts the
assumption on D. If a = 4 or 8, then we have b2D = 28 by (3.12). Thus we obtain
(b, D) = (±2, 7), which correspond to q = 2 ± 7 for α = 4 and q = 4 ± v^-7
fora = 8.
For the case a < 0, we can apply the above argument by replacing a with —a and
obtain (-3 ± V=7)/2, (-1 ± v^7)/2, -2 ± -4 ±

On the values of certain g-hypergeometric series 17
For the case where q is a nonzero imaginary quadratic integer such that det Β n =
0 for some η > 2, we can apply the above argument by replacing q with q~l. This
completes the proof of Theorem 3. •
References
[1] Amou, M., Katsurada, M., Väänänen, Κ., Arithmetical properties of the values of func-
tions satisfying certain functional equations of Poincaré, Acta. Arith., to appear.
[2] Bézivin, J.-P., Indépendance linéaire des valeurs des solutions transcendantes de certaines
équations fonctionnelles, Manuscripta Math. 61 (1988), 103-129.
[3] Duverney, D., Propriétés arithmétiques des solutions de certaines équations fonctionnelles
de Poincaré, J. Théorie des Nombres Bordeaux 8 (1996), 443-447.
[4] Lang, S., Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.
[5] Stihl, Th., Arithmetische Eigenschaften spezieller Heinescher Reihen, Math. Ann. 268
(1984), 21-41.
Addresses of the authors:
Masaaki Amou
Department of Mathematics
Gunma University
1-5-1 Tenjin-cho
Kiryu 376-8515, Japan
E-mail: [email protected]
Masanori Katsurada
Mathematics, Hiyoshi Campus
Keio University
4-1-4 Hiyoshi, Kohoku-ku
Yokohama 223-8521, Japan
E-mail: [email protected]
Keijo Väänänen
Department of Mathematics
University of Oulu
P. O. Box 3000
90014 Oulu, Finland
E-mail: [email protected]

On the Diophantine equation λ^ζγγ = yq
Yann Bugeaud*
Abstract We extend earlier work of Inkeri and solve completely the Diophantine equation
α ^Εγ = yq for 1 < a < χ < 100.
1991 Mathematics Subject Classification: 11D61.
Dedicated to Kustaa Inkeri
1. Introduction
Obláth [6] proved that 4, 8 and 9 are the only numbers which are perfect powers and
whose digits αφ 1 in base ten are identical. This exactly means that the Diophantine
equation
xn — 1
a = yq, in integers η > 3, χ > 2, 1 < a < χ, y > 2, q > 2 (1)
χ — I
has no solution when χ = 10 and αφ 1. Later, Obláth's result was extended by Inkeri
[5] to arbitrary χ e {3,..., 10} and αφ 1, in which case the only solution is given by
4 j^Y = 402. The methods employed in [5] and [6] are based mainly on ingenious
use of classical techniques in the study of Diophantine equations.
The case α = 1 remained open until last year, when Bugeaud & Mignotte solved it
for 2 < χ < 10 [2], and then, jointly with Roy, for χ = ζ' with t > 1 and 11 < ζ < 104
[3,4] : the only three solutions are given by
35 — 1 . 74 — 1 - J 183 — 1 ,
ΤΤΓ = "2· 7TT = 202 -ΠΓΤ = 73·
The author would like to thank the CCCI for providing him a financial support to attend
the Conference in Turku.

20 Yann Bugeaud
Their method rests on various more or less classical techniques in Diophantine approx-
imation, including linear forms in p-adic logarithms [1], together with some heavy
computations.
The aim of the present note is to explain how Inkeri's result can be extended to
other values of the basis x, and to solve (1) completely for χ < 100 and χ = 1000.
2. Statement of the results
We obtain the following extension of Theorem 7 of Inkeri [5].
Theorem. The only solutions of equation (1) with χ < 100 or χ = 1000 are given by
(α, χ, y, n, q) e {(1, 3, 11, 5, 2), (1, 7, 20,4, 2), (4, 7,40,4, 2),
(1,18,7, 3, 3), (7,18,49, 3,2), (7,18,7, 3,4),
(8,18,14, 3, 3), (3, 22, 39, 3,2), (12,22,78, 3,2),
(19, 30, 133, 3, 2), (21,41,1218,4, 2), (13, 68, 247, 3, 2),
(52,68,494, 3,2), (58,99, 7540,4,2)}.
Remarks. All these solutions have been already found by Inkeri, but he did not show
that these are the only ones. Our proof depends considerably on the case a = 1 in
which (1) is already solved. As pointed out by Inkeri [5], "this reveals the decisive
importance of the case a = 1 in the study of the Diophantine equation (1)".
From the results of [4], [3] and of the present note, it turns out that we have at our
disposal an algorithm which enables one to solve (1) for any fixed given value of * ...
provided, of course, that χ is not too large, otherwise the computation would take too
much time.
Our theorem includes the fact that, if a, b and c denote any digits 0,..., 9, then
none of the non-zero numbers
αα.,.αα, abab...abab and abcabc...abcabc
written in basis ten is a perfect power, except, of course, the numbers a, ab and abc
when these are perfect powers.
3. Auxiliary results
In the sequel of this note, we use the following notation. For a prime number ρ and
an integer l non divisible by p, we denote by οτάρ(ί) the least positive integer m such
that im = 1 (mod p).

On the Diophantine equation a——^ =yi 21
The proof of our theorem rests on two elementary lemmas, but, before stating them,
we recall a crucial result concerning equation (1) in the most difficult case a = 1.
Proposition 1. Equation (1) has no solution with a = 1 and χ — ζ', for 2 < ζ < IO4
and t > 1.
Proof. This is the main result of [3]. •
Lemma 1. Let p\,..., p¿ be prime numbers and assume that there are positive
integers u\,..., u¿, η > 3, χ > 2, y > 2, q > 2 with q prime, gcd(q, u\... u¿) = 1
and
pV-pitXi^l = yq· a)
Then there exist a subset S C [l, ...,£}, integers ν > 1 and yi > 2 and a prime Ρ
such that
i'eS
Further, if S is non-empty, then we have, for i e S, that either ordPi (x) is a power of
P, or pi — Ρ andx = 1 (mod P).
Proof. Denote by Ρ the greatest prime factor of n, and write η = Pvm, with
pV
gcd (m, P) = 1. Let w be a prime dividing x xS\l · Then we get xp" = 1 (mod w)
and pnr^j ξ m (mod tu). If w also divides ^wzj, then w is a prime factor of m,
hence is less than m. Thus, we infer from xp" = \ (mod w) that χ = 1 (mod w)
and, since w divides 1 + χ Η 1- xp we get Pv = 0 (mod w), a contradiction.
η ι Pv
We have proved that xPv. and * Τ are coprirne integers. Hence, (3) follows from
χ —1
(2). We then deduce from (3) that for all i e S, we have xp" = 1 (mod /?,), and the
proof of the lemma is complete. •
Lemma 2. Let ρ be a prime number and let ν >2 be an integer. Then we have
J xpV-i xp-l
according as χ = 1 (mod p) or not.
Proof. This lemma belongs to the folklore, but we prefer recalling its proof. Let q be
a prime number dividing ^—γ. We have xp = 1 (mod q) and 1 + xp" + x2pV +
... + χ(.ρ-ΐ)ρν~ι = ρ (mod q), thus gcd(-^^, = 1 if q φ p. Further, we
observe that ^f^ ξ ρ (mod ρ2), and the proof is complete. •

22 Yann Bugeaud
4. Proof
We shall explain our method by treating exhaustively (1) with χ = 100 and
2 < a < 99. Thus, we consider the equation
100" - 1
"ISTΤ-* <4)
in the integers 2 < a < 99, η > 3, y > 2 and q >2, with q prime. Further, there is
no restriction in assuming that a is <?-th power free, and we easily see that a is odd
and not divisible by 5, otherwise (4) cannot be satisfied.
The general strategy of our proof consists in finding a prime number Q which
divides a (100" —1)/(100 — 1) and suchthat Q2 does not, hence (4) cannot be satisfied.
Suppose first that a e {2,..., 99} is prime and apply Lemma 1 to (4). In view of
Proposition 1, the equation
100" - 1
100-1
has no solution (y, n, q) with η > 2 and q > 2. Thus, we deduce that the set S
occurring in the statement of Lemma 1 is equal to {a} and we infer that, if (a, y, n, q)
satisfies (4), then « is a power of a prime and orda (100) is equal to 1 or to a power of
a prime. Hence, only the following 15 values of a remain possible:
a 3 7 11 13 17 19 23 37 41 47 53 59 73 79 83
ordfl(100) 1 3 1 3 8 9 11 3 5 23 13 29 4 13 41
If orda (100) = 1, then η has to be a power of a, since otherwise a divides yq and a2
does not. For the other values of a, we observe that η must be a power of the prime
divisor V of orda (100) and we proceed as follows, setting V = a if orda(100) = 1.
First, if V = 2, we observe that
1002- 1
" löTT
and
by Lemma 2. Thus (4) cannot be satisfied. For the remaining cases, we check that
100v — 1 has a prime divisor Q with Q g {3, 11, a, V} and such that Q2 does not
divide 100v - 1. Hence, by Lemma 2, Q divides a (100" - 1)/(100 - 1) and Q2
does not: equation (4) cannot be satisfied. The following values of Q are suitable:
α 3 7 11 13 19 23 37 41 47 53 59 79 83
Q 7 13 23 7 7 4093 7 271 139 79 3191 53 1231
By exactly the same argument, we deduce that a cannot be a power of a prime. If now
a is composed by two distinct primes p\ and pi, we argue again as above, however,

On the Diophantine equation α^τγ — yq 23
we have to assume that Q & {3,11, p\, p2, V}. We easily check that this can be done:
it suffices to choose Q = 37 if p\ = 7 and P2 = 13.
For χ = 1000, we proceed exactly as above, and we check that for a few number
of primes a we have that orda(1000) is prime or is a power of a prime. For example,
orda (1000) is a power of 2 only for
a e {7,11,13,17,73,97,101,137,193,257,353,449,641,769}.
To conclude, we check that the prime 9901 exactly divides 10004 — 1.
However, a computational problem occurs when
a e S = {173,277,283, 359,439,479, 643, 853},
in which case we, respectively, have orda(1000) = 43, 23,47,179,73,239,107,
71. Using PARI, we see that in each of these cases a||1000ordo(1000) - 1, but, since
the latter number is far too large, we are unable to find another prime factor of
(1000orda(iooo) _ i)/(iooo _ 1). We deduce from (1) that we necessarily have q =
2. To exclude this possibility, it suffices, by Lemma 2, to check that neither a χ
(1000orda(iooo) _ i)/(iooo_ 1) nora χordfl(1000) χ (lOOOortU100°) -1)/(1000-1)
is a square. This verification takes only a few seconds, and we are done. We observe
that the same problem occurs for several values of χ less than 100, and we solve
it as explained above, considering a χ
(jordaM _ 1)/(Λ. _ and α χ orda (Λ;) χ
(xord<1 (x) — 1)/(χ — 1). However, when orda(x) = 1 and when we are unable to find a
suitable prime factor of xord° W — 1, we have to check that neither (xorda (χ> — l)/(x — 1)
nor α χ {χ°*ά<>(χ) — )/(χ — 1) is a square. •
Acknowledgement. The author would like to thank the referee for his careful reading
of the text and his numerous remarks.
References
[1] Bugeaud, Y., Linear forms in p-adic logarithms and the Diophantine equation (χ" — 1)/
(x - 1) = yl, Math. Proc. Cambridge Philos. Soc. 127 (1999), 373-381.
[2] Bugeaud, Y., Mignotte, M., On integers with identical digits, Mathematika, to appear.
[3] Bugeaud, Y., Mignotte, M., Sur l'équation diophantienne (xn — )/{x — 1) = yq, II,
C. R. Acad. Sci. Paris Sér. 1328 (1999), 741-744.
[4] Bugeaud, Y., Mignotte, M., Roy, Y., On the diophantine equation (xn — l)/(x — l) = y9,
Pacific J. Math. 193 (2000), 257-268.
[5] Inkeri, K., On the Diophantine equation a(xn - )/{x - 1) = Acta Ari th. 21 (1972),
299-311.

24 Yann Bugeaud
[6] Oblàth, R., Une propriété des puissances parfaites, Mathesis 65 (1956), 356-364.
Address of the author:
U.F.R. de Mathématiques
Université Louis Pasteur
7 rue René Descartes
F-67084 Strasbourg, France
E-mail: [email protected]

Arithmetical properties of the solutions
of certain functional equations
Peter Bundschuh
Abstract. In sections 0 through 4 the reader will find an extended version of the author's
invited survey talk on the subject in the title which he treated again and again during the
last three decades. Section 5 contains a detailed proof of a new result on the simultaneous
approximation of certain infinite products giving a quantitative version of Theorem 3 below.
1991 Mathematics Subject Classification: 11J72,11J81,11J82.
0. Motivation
When preparing our talk we checked Professor Inkeri's Collected Papers [14] to find
out, if there are some connections with our topic. And indeed, we found three publi-
cations of him, dating from 1960, 1965 and 1976, dealing with irrationality of values
of certain trigonometric functions.
Analytically these investigations depend on the formulae of Hermite which led
him as well as Lindemann and Weierstraß between 1873 and 1885 to their famous
transcendence results on the exponential function. Behind Hermite's analytical for-
mulae is the simple fact, that the exponential function satisfies the differential equation
Df = f.
It is well-known that practically all analytic transcendence methods developed
since Hermite, Lindemann and Weierstraß apply to prove the transcendence of num-
bers which are values of special entire or meromorphic or locally holomorphic func-
tions / satisfying appropriate functional equations, frequently an algebraic differential
equation
F(z, f(z), Df{z),..., Dmf{z)) = 0 (1)
with a polynomial F. The situation concerning analytic methods for algebraic inde-
pendence, or even only for irrationality and linear independence is similar.
As soon as one realizes this fact, it seems quite natural to consider the following
problem. Replace in (1) the differential operator D by the so-called q-difference

26 Peter Bundschuh
operator Aq introduced by Jackson (1908) which is defined by
» ftiz) - /ω
Agf(z) := —
(g- i )z
for fixed q e C \ {1}. Then equation (1) changes into an algebraic g-difference
equation of type
Φ (z,f(z),f(gz),...,f(gmz)) = 0 (2)
where, after clearing denominators, Φ is again a polynomial. As in the classical
situation, one considers quite often the linear case of (2), i.e.
f{qmz) = R0(z)f(gm-lz) + · · · + Rm-i(z)f(z) + Rm(z) (3)
with polynomials Ro,..., Rm. From the analytical point of view, this case was studied
systematically by Poincaré since 1890. Clearly, Aqf tends formally to Df as q 1.
Δ,/(ζ) = /ω « /(Z) = (l + ^—1-)/(-)
1. Irrationality and slightly more
Now, of course, one can ask arithmetical questions concerning numbers which are
values of functions satisfying a functional equation of type (2) or (3) or still more
specifically
f(qz) = Ro(z)f(z) + Rl(z). (4)
Being aware of the fact that in the classical situation the investigation of the exponential
function from the arithmetical point of view led to particularly many and beautiful
results, it is rather natural to start here with a <7-analogue of the exponential function.
To do so, we note that the corresponding functional equation
{g - 1)z>
g >~ vg'
with the initial condition /(0) = 1 is solved exactly by the entire transcendental
function Π;>ι(1 + q(q — l)z), if, for convergence reasons,
we suppose from now on \q\ > 1.
Replacing (q — l)z by ζ in the product, we define
00 00 n
Eq(z) :=Π(1 + ^) = ΣίΓ-Ε
j=l q n=0 Π (qv - 1)
v=l
as q-analogue of the classical exponential function, satisfying the functional equation
Eq{qz) = (1 +z)Eq(z). Historically, this Eq was the second example of a solution of
a first-order Poincaré equation (4), which has been studied arithmetically, by Lototsky
[16] (irrationality).

Arithmetical properties of the solutions of certain functional equations 27
The first such example was
71=0
satisfying Tq(qz) = qzTq{z) +1, cf. Tschakaloff [26]. For his irrationality and linear
independence theorems he used Hermite's classical method of Padé approximations,
whereas Bernstein and Szász [1] got slightly worse irrationality results on Tq using
irregular continued fractions quite similarly as Apéry did in 1978 to prove the irra-
tionality of ζ (3).
Originally, the main interest in the arithmetical investigation of the Tq function
came from its intimate connection with one of the Jacobi theta functions.
The first statement we want to describe more precisely in a moment will be a quite
general arithmetical criterion from which one can easily deduce both irrationality
results, of Tschakaloff and of Lototsky, but also a lot of other later results for which
in each case separate proofs have been given in the past.
The kind of criterion we shall quote is one of so-called Schneider-Lang-Wald-
schmidt type. What do these criteria achieve? Under sufficiently many arithmetical
hypotheses they provide lower bounds for the analytic growth of the transcendental
entire or meromorphic functions involved. Therefore, if the growth of the functions
under consideration is small, they cannot satisfy all arithmetical hypotheses, and in-
deed, this gives the desired arithmetical information on these functions.
Theorem 1. Let Κ be either Q or an imaginary quadratic number field, and let q €
Ok, the ring of integers of Κ. Suppose f(z) = bnzn to be entire transcendental,
and there exist
(i) a sequence (S„)„>o from Kx, with \Bn\ < \q\ß"2+°(>>2) for some real β > 0,
such that Bnbv e Ο κ for ν = 0,... ,η,
(ii) ana e Κχ such that f(aq~m) € Κ, for each m e Ν,
(iii) a sequence (Cm)m>0frorn Kx, with \Cm\ < \q\ym2+o(m2) for some real γ >0,
such that Cmf(aq~ß) € Oxfor μ = 0,..., m.
Then
where \r denotes the maximum of(z)\ on |z| < r.
Idea of proof ([ 5]). One has to choose an appropriate, strongly increasing sequence
(λ/π)»ι>0 of positive integers, and to put zxm := aq~m for m = 0, 1,..., but zx := 0

28 Peter Bundschuh
for all positive integers λ not in (km ). Then we define for i = 0,1,...
Ράζ) :=n<*-a), *:-¿í / fi
λ_1 lfl=*í+i
where empty products are 1, and Rt+\ e M+ is such that all zeroes of P¿+1 are
contained in |£| < R¿+\ . The boundedness of (ζλ)λ>ι is enough to ensure that
oo
f(z) = £ AtPdz)
e=o
holds in the whole complex plane. Since / is not a polynomial, we know Αι φ 0
infinitely often. Calculating the At explicitly via residue theorem, A¿ e Κ becomes
obvious for all £ > 0, by condition (ii). Now the non-zero A¿'s can be estimated
non-trivially from below using the arithmetical hypotheses (i) and (iii). Estimating
\At \ above, using the integral representation in (5), p*(f) comes into play, and from
this we get the asserted lower bound for p*(f) in Theorem 1.
Corollary 1 (essentially Tschakaloff [26, I]). Let Κ and q be as in Theorem 1. If
a e Kx, then Tq(a) $ K.
Very recently, Bézivin [3] proved, by a completely different method (see also
section 2), for integers q 6 Q, that α φ 0 and Tq (a) cannot belong both to a quadratic
number field Κ (maybe real), and this result remains true for certain non-integral
q e Q, depending on the size of 5 := (log 1^21)/ log \q\ |) where q = q\/qi.
Corollary 2 (essentially Lototsky [16]). Let Κ and q be as in Theorem 1. If a €
Κ* \ [~q, ~q2, • • ·}, then Eq{a) i Κ.
η .
Proof. The choice Bn = Π (qv — 1) in Theorem 1 implies β = I. Assuming
v=l
E g (a) e Κ, and putting a = Eq(a) = j with s, t, S,T e Ο κ we find from the
functional equation of Eq
( a\ „Χ"1 Stm m(m+l)/2
' ;·=Λ q ' TT(tqi+s)
7=1
m .
Thus we can take Cm = Τ Π (tq1 + s), and therefore γ = A, and we find p*(E0) >
log'|<?|. On the other hand, p*(Eq) = ^ is well-known, compare part (i) of the
lemma at the beginning of section 5, and we arrived at a contradiction.
Remarks. 1) The huge "difference" in this contradiction can be used to get the
following very good quantitative refinement of Lototsky's result, compare [5] and [22].

Arithmetical properties of the solutions of certain functional equations 29
Theorem 2. Under the hypotheses of Corollary 2 the inequality
Ρ
Eq(a)
|ô|-(5+e)
has at most finitely many solutions ζ with P, Q € OK, Q φ 0, if ε e M+ is arbitrary.
2) With pin) the partition function we have the following identity of Euler (1748)
00 00
Yld-q-jy1 = l + Y(p(n)q-n. (6)
j=l n=1
Since the left-hand side is Eq(—l)-1, Corollary 2 and Theorem 2 give arithmetical
information on the series in (6).
3) In Theorem 1 the lower bound for p*(f) increases, if we include derivatives of
/ at the points aq~m.
4) In Theorem 1 one can take several a's, too. Of this type is a very recent result
of the author [6] concerning the entire function
00
Fq(x, y) := Π*1 + xQ~J + xy<l~2i) (7>
7=1
of two complex variables x, y, introduced by Zhou and Lubinsky [29] (remark
that Fq(z, 0) = Eq(z))·
Theorem 3. Let Κ and q be as in Theorem 1, and suppose α, β e Kx such that 1 ±
aq~i + aßq~2i φ 0 for any j 6 N+. Then the two numbers Fq(a, ß), Fq(—a, —β)
cannot both belong to K.
Zhou and Lubinsky [29], using extensive analytic studies on explicit formulae for
multivariate Padé approximants, could only treat the case α, β e Q+, q e N+. In sec-
tion 5 we shall present a new sharpening of Theorem 3 which studies the simultaneous
approximation of the two numbers by numbers from Κ.
We conclude this section by pointing out the following conjecture which we can
only prove for \q\ > qo(a, β), compare [6].
Conjecture. Fq(a, β) & Κ should be true under the hypotheses of Theorem 3, but
without any assumption on 1 — aq~i + aßq~2i.
2. Linear independence
Using essentially Hermite's analytic method based on the non-vanishing of certain
determinants, Tschakaloff [26, Π] proved the qualitative part of

30 Peter Bundschuh
Theorem 4 (see [7]). Let Κ and q be as in Theorem 1. If a i,... ,a¿ e Κx satisfy
cti/cij $ qz for i φ j (ifí > I), and if ε € R+ is arbitrary, then
\ho + Σ hkTq(ak)\ » (8)
λ=1
holds for any h := (A0,..., he ) e θ£+1 \ {0} with \hx\ < H for λ = 1,..., L
Remarks. 1) The functional equation of Tq shows the necessity of the condition on
cci/aj.
2) For the expression on the left-hand side of (8) the inequality |... | » H~l would
be best possible. The Η-exponent in Theorem 4 is asymptotically 21 for large I.
3) Using the same method as in [7] for Theorem 4, Katsurada [15] generalized our
quantitative result to include now also derivatives, i.e. to estimate
e m
|Ao + Σ ΣΛλμΤ^Μ » Η-1·"\ all |Αλμ| < Η. (9)
λ=1 μ=0
The pure linear independence over Q of
1, Tq{otx),Tq{at),T¡m\aι) T^Hae)
for q, m e Ν and at,..., a¿ e Qx, a¡/aj & qz has been proved much earlier by
Skolem [23] using a quite different method. Skolem's reasoning is an extension of
the method developed by Hilbert (1893) for his transcendence proofs of e and π. This
method - excellently presented in Perron's booklet on irrationality - is much more
arithmetical than Hermite's, and uses essentially divisibility considerations. Using
this Hilbert-Perron-Skolem method, Väänänen and Wallisser [27] gave another proof
of Katsurada's above-mentioned result in the rational case.
Both methods depend on direct constructions of appropriate diophantine approx-
imations. On the contrary, Bézivin [2] developed a new and much more function-
theoretic method for the linear independence of numbers which are values of certain
entire functions generalizing Tq. A central part in this method is played by an appropri-
ate criterion à la Kronecker or Borel-Dwork for the rationality of a suitable auxiliary
function. By the way, Bézivin [3] used this same method for his above-mentioned
recent sharpening of Corollary 1.
We cannot quote here any of Bézivin's linear independence results but we want to
point our that his method doesn't allow, at least until now, to prove quantitative refine-
ments of linear independence. Being aware of this fact, we started with R. Wallisser
a research project, one of the results of which is the following
Theorem 5 (see [11]). Let the following hypotheses be satisfied.

Arithmetical properties of the solutions of certain functional equations 31
— χ
(i) qi,..., qr € <Q are one or several sets of conjugates of integers, multiplica-
tively independent, and one dominating in absolute value,
(ii) there is aprime idealp C OK (where Κ := Q(q\,..qr)) dividing all principal
ideals {q),..., (qr),
(iii) with βι,..., ßr e Qx let A(v) := ßiq\ + • · · + ßrqvr φ 0 for ν e N+,
(iv) for αϊ,..., at e Κχ no quotient a¡/aj with i φ j belongs to the subgroup
{q\,..., qr) of Kx generated by q\,..., qr (if ί > 1 ),
(ν) α,·/Α(υ) € Q for any (i, ν) € {1,... ,i) x N+.
cx> η
Then for the entire function f(z):= Σ ζ"/ Π A(v) the inequality
n= 0 ι>=1
ί
log \h0 + Σ Αλ/(αλ)I » -(log H)2r"r+i)
λ=1
holds for any non-zero h € Ζί+1 with \hk\ < Η for λ = 1,..., ί.
Remarks. 1) If r = 1, then q e Ζ; taking β = l/q gives A(v) = qv thus
n"=l A(v) = qv(v~l)/2 and / = Tq. Since 2r/(r + 1) = 1 we find essentially
Theorem 4 in the form that the expression on the left of (8) is » H~const.
2) We could also include derivatives of / as in [2], [15], [23], [27].
We cannot give here even a sketch of proof for Theorem 5, but from its many
applications to functions / where A( ) is connected with PV-numbers we only quote
the following
Corollary 3. Let (Fn)„>ι denote the Fibonacci sequence 1, 1, 2, 3, 5,..., let S e
Z\{0, ±1}, and suppose that a\,... ,a¿ e Qx have the property that no α,/aj (i φ j)
belongs to (Sg, -Sg~l) C Q(V5)X where g := (1 + \/5)/2. Then for the entire
function
oo
f(z) :=J2zn/(.Sn{n+1)/2F1...Fn)
n=0
the conclusion of Theorem 5 holds.
3. Dimension estimates
Here the problem is the following. Given m e N+, m > 2 and ω = (ωι,..., cúm) e
Rm \ {0}. Find conditions to estimate
DQ(ûj) := dimQ Qoji H (- (Qtom

32 Peter Bundschuh
non-trivially from below. Clearly, Dq(a¿) = m is equivalent with the linear inde-
pendence over Q of ωχ,..., <ym, and in the special case m = 2 we can hope for
irrationality results on ωιΙω\.
We studied this problem jointly with our former PhD student T. Töpfer. The main
result of our efforts was an axiomatization of a method going back to Nesterenko [18]
which is based on linear elimination theory. This axiomatization is so general, and
thus its formulation is so cumbersome, that we don't want to quote it here completely,
all the more as our corresponding paper [8] appeared earlier. One handy consequence
of our main theorem is Nesterenko's
Theorem 6. Suppose ko e N+ and τ\, e R+ with τι > τ2; let φ : N+ ->• R+
be monotonically increasing and unbounded, and let (Aic(X) := λ^\Χι + · • · +
ì-kmXm)k>k0 be a sequence of linear forms over Ζ such that the following conditions
hold:
(i) lim sup (p(k + 1)/<¡í>(&) < 1,
k-* oo
and for any k > ko
(ii) \og{klx + -.. + \lm)<<p(k),
(iii) -τι<p(lc) < log |Λ*(ω)| < -T2(p(k).
Then DQ(©) > (1 + τχ)/{\ + τγ - τ2) (> 1).
IFinstead of (iii), log |Λ^(ω)| ΤΦ(^) ho Ids fork —>• oo, then Dq(o)) > 1 + τ.
Remark. With some necessary, but slight modifications Töpfer [24] could replace
here and in the whole theory Q by an arbitrary algebraic number field.
Jointly with Väänänen we applied Theorem 6 to prove
Theorem 7 (see [9]). Let q e Ζ \ {0, ±1} and a e <Q>X \ {-q, -q2,...}. Then
Dq(Eq(a),E^Ha)) > m(m + l)/(2 m + 6 n~2(m - 1)) for m = 1, 2, 3
(and £)q(. ..)>·· · for m > 4) with sharper estimates for a = — 1. In particular,
Eq (a) and E'q (a) are linearly independent overQ and thus E'q (a)/Eq (a) is irrational.
Our proof used slightly better integrals than
with explicit Rß € Q(a, q), as they appeared in (5), to get an appropriate sequence of
linear forms to which we can apply Theorem 6.
k m—1
lfl=Ä

Arithmetical properties of the solutions of certain functional equations 33
To mention a simple consequence of Theorem 7 we introduce the meromorphic
function
, ,, ν 1
which is a q-analogue of the logarithm.
Corollary 4 (see Borwein [4]). If q, a are as in Theorem 1, then Lq (a) is irrational.
In the special case a — — \,q — 2, 3,... this irrationality was proved by Erdös
[13] much earlier using Lq{— 1) = Σ;>ι(ί?; — l)"1 = Σ„>ι r(n)q~n where τ (η)
denotes the number of (positive) divisors of η 6 N+.
Whereas Erdös' method doesn't lead to quantitative statements on irrationality, i.e.
to irrationality measures, our main result with Töpfer (from [8]) is general enough to
include even measures for the linear independence over <Q> of the earlier ωι,..., <um if
DQ(O¿) — m. As an application to Lq(a) we state, under the hypotheses of Theorem
7 and Corollary 4,
\ho + hxLq(a)\ » Η-3·310", |A0 + hxLq{-1)| » //-1·508···,
compare [9]. Very recently Matala-aho and Väänänen [17] sharpened the first estimate
to » H~2·946· · using, essentially, more appropriate linear forms Λk than we did in [9].
4. Transcendence
Until now we didn't discuss much more than irrationality questions. But it is very
natural (and not hard) to propose open transcendence problems in the present field,
too. Here are two:
Suppose q € Q, \q\ > 1, anda € QX. Is it true that Tq(a) is transcendental? Or
that Eq (a) is either zero or transcendental?
Of course, the answer should be "yes" in both cases. What one would try, first of all,
is to use the classical transcendence methods of Gelfond and Schneider, respectively.
Since bothapply essentially to entire or meromorphic functions, one should work with
fixed q e Q, and look at Tq(z), say, as an entire function of ζ as we did before.
a) Gelfond's method. Here we would need an algebraic differential equation for
Tq. But such an equation can definitely not exist, by the following theorem due to
Popken [21]:
If an entire function f{z) = Σ„>ο bnzn with all bn e Q satisfies an algebraic differ-
ential equation, then b„ φ 0 implies — log n\ <&n log2 n. Clearly, bn = q~n(n~1)/2
in the case of f = Tq shows that this function is certainly hypertranscendental.

34 Peter Bundschuh
b) Schneider's method. Here, again definite transcendence results could not be proved
until now. What one can get via Schneider's method are theorems of the following
weak type, compare [10]:
Let Κ be an algebraic number field, q e Κ, and suppose RQ, RI € K[X] in the
functional equation (4) for the entire transcendental function f. Then we have for
any d € N+
card{a e Q | |q\~l < \a\ < 1, f(a) 6 Q, [K(a, f(a)) : K] < d)
< 700D[K : Q] P^-d2,
log Iii
where D denotes the degree of RQ, and h(.) the absolute logarithmic height.
A similar result, including now derivatives, can be proved using Gelfond's method
from a).
c) Mahler's method. About seventy years ago, Mahler developed a method to establish
the transcendence of numbers which are values, at certain algebraic points, of func-
tions of several variables satisfying a class of functional equations. We describe the
main hypotheses for this method very roughly to show its connection with the above
transcendence problem concerning Tq(a), say.
If Ω = (ω,·,·) e Mat(í χ /; Ν) and ζ := (zi, • • •, Zt), then define
Suppose that / : U C, U an open neighborhood of 0 € C, is holomorphic,
has Taylor coefficients about 0 in some fixed algebraic number field, and satisfies a
functional equation of type
/(ΩΖ) = (¿ a^zìfizA/(¿ bß(z)f(zA
μ=0 ' V=0 '
with αμ, bß € C[z], not both of am, bm zero, and 1 < m < γ(Ω). Here γ(Ω) denotes
the maximum of the absolute values of all eigenvalues of Ω, and if an eigenvalue λ has
|λ| = γ(Ω), then λ = γ(Ω). Then, under a minor, but very natural further condition,
one has f(a) g Q for all a e U Π Q* with ai... at Φ 0.
Let us now look at f(zi,zi) := z\z^n~Y)l1 in C χ t/i(0) =: U where
C/i(0) := {Z2 e C | |z2| < 1}· We have
00
Zlf(ZlZ2, Z2> = Σ A+l4n+l)ß = m,Z2) - I-
n=0

Arithmetical properties of the solutions of certain functional equations 35
Therefore this / satisfies /(Ωζ) = (/(ζ) — l)/zi, and we have m = 1,
!)·
and thus Γ(Ω) = 1, unfortunately. Otherwise we could conclude^ (a) = f(a,q~l) g
Q for any a e Q*,q e Q (|g| > 1), and this would prove exactly the above conjec-
ture.
Since about thirty years there was a big progress with Mahler's method, compare
[20], but the condition r (Ω) > m resisted to all attacks, at least until now.
In a letter exchange on the problem of Tq(a) we had in the early 1980's with
Professor Mahler, he wrote (in German): "In spite of many investigations, the tran-
scendence of theta functions remains unsolved, and thus every partial result is of great
interest." And he continues: "... and I guess that the transcendence of Tq (a) will not
be solved before the next century".
d) Nesterenko's recent progress. Nevertheless, a very big progress in this area was
made by Nesterenko [19] in 1996 who proved even algebraic independence results for
certain numbers related to modular functions. We describe one of his main results:
Let
OO 00
P(z) := 1 - 24 σι(n)zn, Q(z) := 1 + 240^] σ3(η)ζ",
71=1 n=1
oo
R(z) := 1 - 504 σ5(η)ζη
n=1
in i/i(0), where Oj(n) := lú¿\ndK Then for any q e C, \q\ > 1 the transcendence
degree of the field Q(g, P(~), ß(±), /?( j)) over Q is at least 3. This implies the
algebraic independence of Σ„>ι aj(n)q~n, j = 1,2,3, for any q e Q, \q\ > 1.
Shortly later, Duverney, the two Nishiokas, and Shiokawa [12] deduced from
Nesterenko's result the transcendence of
OO 00 00
Yjq-n(n~X)l2{=Tq{)), £ <T"2(= 7^2(1/4)), Π*1-«""^^-1»
n=Ο η- 0 n=1
and of j p(n)q~n for any q e Q, \q\ > 1.
5. A quantitative version of Theorem 3
The main aim of this section is to present a complete proof of the following statement.
Theorem 3'. Let K, Ok, q, α, β be as in Theorem 3, and let Fq denote the infinite
product defined by (7). Then there exist effectively computable γ\, γ2 € R+, depending

36 Peter Bundschuh
at most on q, a, ß, such that for any {P\, P2, Q) e θ\ with Q Φ 0 the following
inequality holds
max(\QFq(a, β) — Pi|, \QFq(-a, -β) - P2) > yi|ßr6-w<Ioelßl)"l/2.
In the proof we need the following auxiliary result of Töpfer, compare Lemmas 2,3
and 5 of [25], on solutions of the above functional equation (4): f(qz) = RO(z)f(z) +
RI(z) with polynomials RQ, R\.
Lemma. Suppose that f (ζ) = Έ^_0όηζη is an entire transcendental function satis-
fying (4).
(i) Then for all sufficiently large r e R+ the asymptotic equation
log\r = Pi log2 r + pi logr + 0(1)
holds with pi := t»/(21og pi := (log |aol)/(log \q) — ω/2, where ω and
ao denote the degree and the leading coefficient of RQ, respectively.
(ii) Furthermore, there exists a constant ci e R+ such that the inequality
log n\ < —(λι«2 + λ2η — ci)log 1^1
holds for any η e Ν, where λ ι := 1/(2ω), λ2 := —ρι/ω.
(iii) Finally, there exists a constant c2 e M+ such that for any sufficiently large η e Ν
the inequality
log j\ > -aij2 + k2j + C2) log \q
holds for at least one j € {n + 1,..., η + ω}.
To prove Theorem 3' we have to apply the preceding lemma to f(z) :=
Fq(az, βζ) which satisfies functional equation (4) with Ro(z) :— 1 + aqz + otßq2z2
and /?i(z) := 0. Thus we have to apply the lemma with ω = 2, ao = aßq2 (by
αβ φ 0), pi = l/(log |i|), pi = (log \aßq)/(\o% |^|) (we denote this expression by
ρ), λι = 1 /4, λ2 = —p/2. By (ii), the Taylor coefficients bt of our function / about
the origin satisfy
t\ < \qne2+$e+Cl (10)
for any Í e Ν, and, as soon as Í is large, we have
j\ > Iqriï+îj-* (11)
for at least one j e {ί + 1, ί + 2}.
Subsequently we consider the integral
„ , 1 [
Hmo'n):=2¿l J Γ^Ί ^ (12>
If \=R ro+1 · Π {ζ - q~v) · Π (ξ + q-v)
υ=0 v=0

Arithmetical properties of the solutions of certain functional equations 37
with mo e Ν, η e N+ and a real number R > 1. Integrals of this type appeared yet
in the proof of Theorem 3 (compare formula (11) in [6]). With Ν := mo + 2η and
Ρη(ζ) := n"~¿ (1 - ζ-2ς~2ν) we have the trivial identity
1 f fttW • 1 f 1 - pn(0
/(mo,«):=— J + — J ^ · ,dit (13)
lfl=* lfl=*
leading to
I(mo,n) = bN + J(mo,n), (14)
if J (mo, n) is defined by the second integral on the right-hand side of (13).
Now we assert (see (16) and (17) below), that J (mo, n) is "often small" compared
to bf] as Ν -»• σο. Namely, if zo, • · ·, zn-\ are arbitrary complex numbers, then
η-1
1 ~ Π(1 ~Ζν) =1 -11 - Σ Σ zv,zv2-+·•·}
v=0 0<ui<n 0<vi<v2<«
=έ<-ΐ)*+ι Σ
jfc=l 0<vi <—<Vk<n
Clearly, we have the estimate
η-1 k
Σ Zv, ...Zi* I < ^IZvJ.-.IZvJ < (Σ|ΖυΙ) *
0<ι>ι <—<vic<n ••• v=0
Using the above definition of Ρη(ζ), these two remarks lead, for all complex ζ with
\ζ\ = R,to
H - Pn(OI < έ(Σ*"2Ι«Γ2ν)* < ¿(*2( 1 -1q\-2)Yk
k=\ v=0 k=l
<(R2(\-\q\~2)-
andhere the right-hand side is at most 1/2, if we suppose Ä2(l —\q\~~2) > 3. Therefore,
under this condition on R, we have \Ρη(ζ)\ > 1 /2 on |ÇI = R from which, using the
integral definition of 7(mo, η) in (13) and (i) of the lemma, we estimate
\J(m0,n)\ < c3|/|Ä/r(JV+2) <c4expQ|^-(W + 2-p)log/^ (15)
where, here and later, all c¡ are positive real constants depending at most on q, α, β.
The right-hand side of (15) becomes minimal for R = \q\(N+2~p)/2, and this choice
of R (ensuring also R2( 1 — \q\~2) > 3 if TV is large enough) leads to
\J(mo,n)\<c4\q\-^+2-^4 = c5\qnN2^N-N. (16)

38 Peter Bundschuh
For large m € Ν we consider now the two integrals I (mo, η) with mo = m — 1
and mo = m, respectively. In the first case we have Ν = m + 2n — 1, in the second
Ν = m + 2n, and, by (11), we know
N\ > I(17)
for at least one of the two just mentioned Ν. Therefore we have for at least one
mo e {m — 1, m)
/(mo, η) =bJ\ + = M1 + 0(\qrN))
V bff J (18)
= \q\-iN2HN+°w,
by (14), (16), (17), and (10).
Similarly to the formulae (12), (13), (14) in [6], we calculate from (12), by the
residue theorem, the following formula
q-n(n-l )/(mn)= y^ (.jjb+S^V, ο^,+Σ^νίσ,,+σ,,+ν) («• ß)
σ2η
σ0+···+σ2η=πι Π (qK ~ 1)
K=l
qv(m+2v)
v=0 ~ri(q2i + aqJ + aß) • Π (1 - q2lc) · " Π \qhc ~ 1)
7=0 κ=1 κ=
(19)
(_] \m n 1 „v(m+2v)
+ -β) Y — q- .
2 r'¿Jv-l , υ . η—ν—I „
ν=0 Π (q2J - aqJ +αβ)· Π (1 - q2lc) · Π (q2lc - 1)
j= 0 ΑΓ=1 κ=1
Here the ca (α, β) e Κ are given by baq~a (qK -1) for any σ e N, see Lemma 4
in [6]; there the reader will find their most important properties.
As we have seen earlier, compare (18), for any (m,n) € N+ with sufficiently large
m + η, we can find a Λ = A (m, n) e {/(m — 1, η), I (m, «)} satisfying
|Λ| = |9|-|('"+2")2+0(m+n) (20)
On the other hand, (19) can be written as
A = a(m, n) + Fq(a, ß)b(m, η) + Fq(-a, -ß)c(m, η) (21)
with obviously defined a, b, c e Κ for which we estimate directly from (19)
\, |c| < \q\'""+"2+0(m+n) (22)
Next, for the a, b, c in (21) we try to find a common denominator, i.e. an Η =
H(m,n) e Kx such that all
A := A(m,n) := Η-a, Β := B(m,n) := H b, C := C(m, n) := H c (23)

Arithmetical properties of the solutions of certain functional equations 39
are in Or- From (19) we see that it is enough to take the product
n—2 η-1
q-n(n-1) 2ym Y[ Y(q2j + aqj + aß) · [] y(q2j - aqj + aß) • V(m, n) (24)
;=0 7=0
for H, where γ e Ο κ \ {0} satisfies γα, γαβ e Ο κ and Vim, ή) e Ο κ \ {0} is a
common multiple of the following two products from Ok
m η—1
π «--υ. Π^-1)
κ=1 K=l
Clearly, for any m > 2n — 2 the second product divides the first in Οχ- Therefore we
can apply the preceding considerations with m = 2n — 1, and we can finally take H
to be the number (24) with V(2n -I ,n):= Π^1 (qK - 1) leading to
\H\ = \q?n2+0(n\ (25)
Putting L := HA with this Η and with Λ = A(2η — 1, η) we get from (20) and
(25)
\L\ = I q\-n2+°™. (26)
Furthermore, we find
L = A + Fq(a,ß)B + Fq(-a,-ß)C (27)
with the Λ = A(2n — l,m),... € 0K from (23) for which, by (22) and (25), we
know the upper bounds
\C\ < \q\6n2+°M. (28)
Next, we define an unbounded sequence (φη ) of real numbers, strictly increasing
from some point on, by φη := |q C6". Here we choose C(¡ e IR+ in such a way that
the inequality
\<Pn+\L\ < Iq\-" (29)
holds for all large η e Ν, compare (26). On the other hand, by an appropriate choice
of C7 € R+, we have
\<PnL\ > \q\~cin. (30)
Now we suppose Q e Οχ given, sufficiently large in absolute value, and we determine
ti = n(Q) e N+ uniquely such that
ψη — \ Q\ < ψη+ϊ (31)
holds. This is equivalent with
n2 - c6n < < n2 + (2 - c6)n + (1 - c6). (32)
log I? I

40 Peter Bundschuh
Denoting ej := QFq(a, β) - P\, ε2 := QFq(-a, -β) - P2 with arbitrary
Pi, P2 e OK, the hypothesis
max(|ei|,|£2|)< Iii-6"2"08" (33)
with the above η = η(β) and with an appropriate e E+ would imply \eiB+e2C\ <
Iq\-", by (28). By (27), (31) and (29)
IQA + (Λ + εχ)Β + (P2 + s2)C\ = \QL\ < \<pn+xL\ < ^Γ" (34)
such that I QA + P\B + P2C\ < 2\q\~n would hold, implying QA + Λ Β + P2C = 0,
since η = n(Q) is large, by (31). Using this last equation in the left part of (34), we
find
\eiB + e2C\ = \QL\ > \<pnL\ > \q\~cin,
by (31) and (30). Bounding above the left-hand side of this inequality, using (28), we
conclude that we have in any case (compare the contrary of (33))
maxa^U^D^M-6"2-'9". (35)
This estimate, combined with the inequalities (32) and with the above definition of
the s¡, proves Theorem 3' completely.
Remark. If one doesn't yet fix m before (25), but one takes m — [τη] with some real
τ > 2 to be determined later, then the numerical factors of n2 in the formulae (25),
(26), (28), (32), (33), (35) depend on τ. This dependence is such that the exponent 6
in the inequality of Theorem 3' changes into φ(τ) := 2(r2 + 2r +4)/(4r — r2) under
the condition that we take care for τ < 4. Plainly, we have φ(τ) > 6 for 2 < r <4
with equality if and only if τ = 2.
References
[1] Bernstein, F., Szász, O., Über Irrationalität unendlicher Kettenbrüche mit einer Anwen-
dung auf die Reihe , Math. Ann. 76 (1915), 295-300.
[2] Bézivin, J.-P, Indépendance linéaire des valeurs des solutions transcendantes de cer-
taines équations fonctionnelles. I, ManuscriptaMath. 61 (1988), 103-129; II, Acta Arith.
55 (1990), 233-240.
[3] —, Sur les propriétés arithmétiques d'une fonction entière, Math. Nachr. 190 (1998),
31-42.
[4] Borwein, P. B„ On the irrationality of Σ(1/(ςη + r)), J. Number Theory 37 (1991),
253-259.
[5] Bundschuh, P., Ein Satz über ganze Funktionen und Irrationalitätsaussagen, Invent.
Math. 9 (1970), 175-184.
[6] —, Again on the irrationality of a certain infinite product, Analysis 19 (1999), 93-101.

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Title: Tunnustus: Novelli
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Produced by Juhani Kärkkäinen and Tapio Riikonen
TUNNUSTUS
Novelli
Kirj.
MAKSIN GORKI
Suom. Hella Wuolijoki
Turku, Sosialistin Kirjapaino-Osuuskunta I. L. 1908.

SUOMENTAJAN ESIPUHE.
Katso:
"… Kaikkia teitä ja polkuja myöten kulkee horjuen harmaita
olentoja, pussit selässä ja kepit kädessä; he kulkevat ilman kiirettä,
vaan kumminkin joutuisasti, päät syvästi alaspainuneina; he kulkevat
lempeinä, miettiväisinä, luottavaisin, avonaisin sydämin…"
"Heitä kulkee, kulkee nuoria ja vanhoja, naisia ja lapsia, ikäänkuin
yksi ja sama ääni olisi kutsunut kaikkia ja tuntuu jonkinlainen
ääretön voima siinä loppumattomassa vaelluksessa maan kaikkia
teitä myöten, ja se voima valtaa minut, saattaa sieluni kuohumaan,
ikäänkuin se lupaisi jotain sielulleni?…"
— Mene, kysy, tiedustele!
"… Tuossa ihminen kulkee tottelevaisena, varovaisena; se hakee,
katselee, kuuntelee herkkänä ja taas kulkee, vaeltaa. Maa soi
hakijain jalkojen alla ja työntää heitä etemmäs, jokien ja mäkien,
metsien ja meren kautta, — yhä kauvemmas, kaikkialle sinne, missä
seisoo yksinäisiä luostareita, jotka lupaavat ihmeitä, kaikkialle sinne
missä hengittää toivo johonkin muuhun kuin tähän katkeraan,
vaikeaan, ahtaaseen elämään."

Niin kulkee Venäjän kansa, talonpoikaiskansa, hakemassa jumalaa.
Elämä on tullut levottomaksi, sietämättömäksi; ihmiset ovat
kadottaneet tuon vanhan jumalan, joka heillä oli vanhoina,
rauhallisina aikoina. Nyt he hakevat. Ja ennen kaikkea vanhoista
luostareista, kirkoista.
Löytävätkö he sen sieltä?
Maksim Gorjki tunnustaa tässä kirjassa oman
jumalanhakemisensa.
Se on tarina jumalanhakijasta, joka löytää jumalansa kaikkien
jumalien synnyttäjästä — kansasta.
Kirja on ensimäinen laadultaan, tarina siitä miten ajatteleva,
syvästi tunteva ihminen taistelee itsensä personallisen jumalan
uskonnosta köyhälistön maailmankatsantokantaan. Ihminen
taistelee, kärsii äärettömiä sieluntuskia ja pelkoa, hyläten
kaikkivaltiaan, auttavan, palkitsevan jumalakäsitteen sielustaan, sen
jumalakäsitteen, jota se ei huomaa missään, vaan jonka
olemassaolon edellytys on ihmisen turvana, — ja ihminen voittaa
olemattoman edellytyksen sijaan avun elämälleen, turvan ja
lohduttajan — uudesta jumalastansa, kansasta, tovereistaan.
Sellaisesta ihmisestä kirjoittaa Gorjki.
* * * * *
Jumalanhakijoita on koko maailma täynnä, on niitä Suomessakin.
Sellaisia hiljaisia, syviä, miettiväisiä ihmisiä, jotka tahtovat oikeutta
ja hyvää ja kärsivät siitä, että he itse eivät aina kykene

menettelemään oikein ja hyvin, tahi sen mukaan kuin heille on
opetettu.
Hiljasilla metsäseuduilla lienee ihmisiä, jotka ennen muinoin elivät,
työskentelivät, maksoivat veronsa, kuolivat, uskoen vanhaan
jumalaan ja taivaaseen kuoleman jälkeen. Nyt ovat taloudelliset olot
muuttumaisillaan, ajat ovat tulleet levottomiksi, maailman elämä
koskee metsäseutujen asukkaisiinkin, torppari pelkää häätöjä,
mökkiläinen työttömyyttä. Jostain kaupungista ilmestyy ihmisiä, jotka
sanovat, että työväki siellä on laittanut puolueen, joka tappelee
työtätekevän kansan etujen vuoksi, joka tahtoo, että kaikki saisivat
pitää torppansa, mökkinsä, että työväki tahtoo parempaa palkkaa,
työtaakkaa vähennetyksi ja muita elinehtojaan parannetuiksi.
Hiljainen ihminen metsäseudulla tuntee, että nämä ihmiset ajavat
sen etua, liittyy niihin joko saavuttaakseen itse etuja ja suojaa tahi
helpottaakseen lastensa elämää, jotka maattomina kulkevat työtä
hakemassa.
Ja elämä tulee helpommaksi, valoisammaksi.
Mutta mitenkäs on uskonnon laita?
Toiset siellä sanovat, että nämä uudet ihmiset, jotka tappelevat
kansan puolesta, tahtovat hävittää jumalan ja uskonnon; ja mitä
ihminen sitte muka tekee kuoleman sattuessa?
Lorua, — kyllä metsäseudun mieskin tuntee, että nämä uudet
ihmiset täyttävät paljoa paremmin rakkauden ja oikeuden ja
puhtauden käskyt, he auttavat toisiaan ei ainoastaan sanoilla vaan
teoilla, auttavat häntäkin, paremminkin kuin niiden vanhojen
ihmisten jumala sitä on tehnyt. Mutta kuitenkin. Täytyy ajatella

loppuun saakka: mikä tulee kuolemani jälkeen, onko siellä jumala,
onko taivas? Mihin minä menen?
Uudet ihmiset tahtovat laittaa taivaan maan päälle, tahtovat että
kaikilla ihmisillä maan päällä olisi vielä parempi kuin tuolla pappien
lupaamassa taivaassa. Tuntuu siltä kuin ei jumalaa eikä taivasta enää
tarvittaisikaan…
* * * * *
Tässä "Tunnustuksessa" sanoo vanha lukkari:
"Jumala on sitä varten, ettei ihmistä pelottaisi kuolla, vaan miten
elät, se on sinun asiasi."
Jumalanhakija kulkee elämässä ja kyselee kaikilta minkälainen
jumala kullakin on ja hän huomaa, että ihmiset elämässään eivät
todellakaan enää tarvitse suurta, kaikille yhteistä jumalaa. Joka
ihmisellä on oma jumalansa.
"Ihmiset ovat hajottaneet jumalan palasiin, jokainen tarpeensa
mukaan."
Yhden jumala on puolittain lääkäri, toinen taas pitää jumalasta
kiinni, koska pelkää kuolemaa; kolmannen jumala on hyvä,
neljännen — hirmuinen. Papit ovat palkanneet sen rengikseen ja
maksavat sille palkan rukouksilla ja pyhällä savulla.
Jumalaa komennetaan rankaisemaan ja palkitsemaan, jumalan
täytyy ennättää täyttämään ihmisten mitättömät, pienet rukoukset.
Varas rukoilee että jumala auttaisi häntä varastamaan toisen
omaisuutta, koska varkaan on nälkä. Omaisuuden omistaja taas

rukoilee, että jumala suojelisi hänen omaisuuttansa varkaalta.
Niin! Elämässä ihmiset tulevat toimeen ilman jumalaa, he
tarvitsevat ainoastaan palvelijaa, joka tekisi kaikkensa heidän
hyvinvointiaan varten ja joka on erilainen jokaisen mielestä.
Ymmärrettävää on, ettei totinen ihminen voi palvella sellaista
pientä jumalaa.
Entäs kuolema?
Niin, siinä ihmiset tarvitsevat jumalaa. Ihminen ei tahdo lakata olla
olemassa.
Ja kuitenkin hän näkee, miten toiset ihmiset nauravat, iloitsevat,
surevat ja kärsivät, miten he kuolevat kuin kärpäset, häviävät tuosta
aurinkoisesta, säteilevästä maailmasta.
Kysymys jumalasta on sama kuin kysymys kuolemasta.
Ja ketkä kysyvät: mitä tulee kuoleman jälkeen, mistä ihminen on
tullut, mihin hän menee?
Juuri ne, joiden elämä on tullut siksi levottomaksi, että usko siihen
kaikkiauttavaiseen jumalaan on lakannut, usko siihen, ettei ihmisen
pitäisi huolehtia nykyelämän suhteen, koska taivaassa kerran kaikki
kärsimykset palkitaan. Juuri ne, jotka tuntevat itsensä onnettomiksi,
turvattomiksi maailmassa ja joilta kokemus on hävittänyt uskon
kaikkivaltiaaseen jumalaan.
Siellä, missä kuten Venäjän talonpoikien keskuudessa, vuosisatain
kuluessa taloudellinen elämä on ollut pysähtynyttä, jähmettynyttä,
siellä ovat aina kehityksen ja epäilyksen kannattajina ja levittäjinä

olleet sellaiset ihmiset, jotka jostain syystä ovat joutuneet
ulkopuolelle yhteiskuntaa, joko rikkomalla tuon pysähtyneen
yhteiskunnan tapoja vastaan tahi koska heidän taloudellinen
asemansa on eronnut ympäristön taloudellisesta asemasta. Sellaiset
ulkopuolella yhteiskuntaa "pohjalla" olevat ihmiset, juopot, rentut,
paljasjalkaset "jumalan kukkaset" ovat Gorjkin lempilapsia, koska
oikeastaan, niin ihmeelliseltä kun se kuuluukin, juuri nämät ovat
Venäjän kukkasia, Venäjän liikkuvin aines, täynnä kiukkua tuota
harmaata, likaista elämää vastaan, jossa kansan joukot makaavat
sortavan virkavallan nukuttamina.
Juuri niin, koska sellaisen, luokkansa ulkopuolelle joutuneen
ihmisen taloudellinen asema epävarmuutensa ja tavattomuutensa
takia herättää epäilyksiä maailmanjärjestelmän suhteen. Sellainen
epäilevä ja jumalaa hakeva ihminen on myöskin tämän kertomuksen
sankari, ja sitä ovat muutkin huomattavimmat henkilöt, jotka siinä
esiintyvät.
Kertomuksen päähenkilö on avioton lapsi, löytölapsi. Jo tämä
hänen erikoisasemansa sellaisessa talonpoikaisyhteiskunnassa antaa
hänen ajatuksilleen erikoisleiman, siihen suuntaan vaikuttavat vielä
hänen kasvattajansa, juoppo, yksinäinen luonnonihminen, lukkari
Larion ja tunnettu hevosvaras, leikinlaskija ja laulaja Savelka.
Sellaisia olentoja on kertomuksessa useita ja kertoja Matvei on itse
vallan samanlainen. Hänen erikoisasemansa panee hänet
ajattelemaan ja hakemaan jumalaa, sellaista jumalaa, joka
todenteolla olisi olemassa, joka voisi ja tahtoisi auttaa ihmisiä,
sellaista eheätä, yleispätevää jumalaa, joka ei olisi pappien palvelija.
Pitkä on jumalanhakijan matka ja paljon ihmiskunnan tuskia ja
suruja näkee hän sillä matkallaan, mutta samalla hän huomaa, ettei

hän ole yksin, että paljon on maailmassa sellaisia hakijoita kuin hän,
ja silloin hän saa käsityksen kansasta. Hän kuljeskelee luostareita
pitkin, yhden erakon ja viisaan jumalanmiehen luota toisen luo ja
huomaa, että useat itsekin epäilevät jumalan olemassaoloa, tahi ovat
ainoastaan uhraamalla järkensä päässeet noista epäilyksistä. Ja se
mies taas, joka nähtävästi on joutunut elämän huipulle, ainoa joka ei
tuskastu jumalanhakijan kysymyksiin, kaunis, viisas, sivistynyt,
nautinnonhaluinen munkki Antonius, myöntää kysyjälle vasten
naamaa: "ainoastaan ihminen on olemassa, kaikki muu on vaan sen
mielipidettä". Ja tämä tappaa Matvein uskon siihen olentoon, jonka
sanottiin olevan ulkopuolella ihmistä ja ohjaavan ihmisen kohtaloa.
Tästä lähtien tarkastaa jumalanhakija enimmäkseen juuri ihmisten
kohtaloita ja miten eri kohtalot eri ihmisille ovat laittaneet eri
jumalat, eri käsitteet jumalasta ja sitten hän tulee siihen
johtopäätökseen, ettei ole olemassa sitä suurta jumalaa, joka hänet
lapsena rukouksen aikana yhdisti koko suureen luontoon. "Ihmiset
ovat repineet tuon lapsen jumalan kappaleiksi, jokainen tarpeensa
mukaan."
Siis pääpaino tässä maailmassa on ihmisissä, eikä jumalassa.
Ja sitte kun maailma on opettanut Gorjkin jumalanhakijan
katselemaan ihmisiä, tutkistelemaan niitä, niin löytää hän
vaellusmatkallaan ihmisen, samanlaisen luokastaan ulospotkitun
ihmisraukan, joka puolestaan on löytänyt jumalansa ja joka ilmaisee
sen tällekin hakijalle.
"Kuka on jumalanrakentaja?" kyselee Matvei.
— Kansa se on! Lukematon maailman kansa! Suurempi marttyyri
kuin kaikki ne, joita kirkko on pyhiksi tunnustanut! Kuolematon

kansa… ja hän on kaikkien, niin entisten kuin nykyistenkin jumalien
isä!
Ihmiskunta se on, joka on aina luonut jumalakäsitteensä sen
mukaan kun se on tarvinnut niitä kehityksessään ja ihmiskunta se
on, joka ne kerran hävittää, silloin kun se niitä enään ei tarvitse.
Matvei on jo nähnyt vaelluksensa aikana, että ihmiset itse
muodostavat jumalansa ja nyt se lausutaan julki.
Kummallinen viha puhuu siitä vanhasta poispotkitusta papista,
joka on tullut sosialidemokraatiksi ja taistelee salaista, vaan varmaa
ja lakkaamatonta taistelua tuon jumalanluojan, kansan edestä! Hän
vihaa tuota ihmisen yksinäistä, omia etujaan valvovaa "minää", joka
on repeytynyt pois äidistään, kansasta, joka vapisee pelosta
yksinäisyytensä takia ja joka turvatakseen itseään ja omaa
hyvinvointiaan murhaa muita ja luo jumalia, jotta nämä palkitsisivat
sitä jossain taivaassa, silloin kun elämä maailmassa tuottaa
kärsimyksiä.
Ja tämä kulkuripappi jumaloi kansaa ja sen kärsimyksiä, siitä
hänen vihiinsä yksilöllistä minää vastaan, joka on eronnut kansasta,
jumalasta ja jumalanluojasta, on sortanut kansanjoukkoja, on luonut
kaikki onnettomuudet maailmaan. Hän kertoo Matveille sorretun
ihmiskunnan historiaa ja Venäjän kansan kohtaloita, mutta koska
hän itse on voimakas taisteluluonne, joka kaikin elinvoiminensa
tappelee tuota minäänsä vastaan, niin muodostuu hänen
käsityksensä maailmasta ja sen kehityksestä vallan uudeksi
uskonnoksi. Hän ei huomaa enää mitään siitä, miten paljon hyötyä
tuo yksilöllinen minä on tuonut maailmaan. Hän vaan vihaa sitä,
koska se nykyään on ihmiskunnan kehityksen tiellä, tuon jumaloidun
kansan tiellä.

Matvei on ihan kuin huumauksessa, hän ei uskalla eikä kykene
suostumaan tuohon uuteen oppiin, jonka hän jo kauvan on
aavistanut. Pappi lähettää hänet erääseen tehtaaseen työläisten,
uutten ihmisten luokse, näkemään omin silmin, että tosiaankin uusi
ihmislaji on maailmassa muodostumaisillaan.
Mestarillisesti on Gorjki kuvannut Venäjän herääviä
työläistyyppejä.
Jumalanhakija saapuu heidän luokseen kysymyksineen: "Missä on
Jumala?
Mistä ihminen tulee, mihin menee?"
Ennen kaikkea karsitaan tuolta hakijalta pois käsitys hänen
erityisestä "minästään", osotetaan, että sellaisia kuin hän on,
miljooneja maailmassa, jotka joko katselevat kipuineen maailmaa
jokainen omasta luolastaan ja itkeä nyyhkyttävät tämän onnettoman
maailman takia, tahi jos niillä on tilaisuus, niin ratsastavat toisten
selässä, nylkevät muita ihmisiä.
Ja kun jumalanhakija on mielestänsä tullut tavalliseksi
joukkoihmiseksi, silloin saa hän vastauksia kysymyksiinsä ja saa niitä
kahdelta eri taholta.
Työläisissä siinä tehtaassa erottautuu kaksi eri tyyppiä jotka
antavat tavallaan eri vastauksia niihin kysymyksiin.
Setä Pjotr, voimakas, taisteluhaluinen seppä, joka kokonaan on
kiinni köyhälistön taistelussa, ei tarvitse jumalia eikä sellaisia
kysymyksiä ollenkaan:

"… Ei ole jumalia! Uskonto, kirkkoja muu sellainen on kuin synkkä
metsä; se on synkkä metsä ja siinä asuu ryöväreitä! Petosta vaan
kaikki!"
Hän on työläinen, joka luokkavaistonsa kautta on löytänyt
tehtävänsä elämässä, joka elää vaistonsa kautta luokkansa elämää,
taistelee itsetietoisesti sen edestä, ei erota itseänsä eikä omia
etujaan luokkansa eduista ja elämästä. Hän ei ajattele kuolemaa, ei
tunne sitä, ei välitä elämän alusta eikä lopusta, kuten nuori
tanssijapari, poika ja tyttö, jotka yhdessä sopusointuisesti,
voimakkaasti liikkuen nauttivat musiikista, liikunnasta,
sulautumisesta toisiinsa, välittämättä, tietämättä, että heitä koskaan
maailmassa odottaa kuolema tahi jonkinlainen onnettomuus. Hän
vihaa jumalan nimeä ja käsitettä, eikä kärsi, että siitä ollenkaan
puhutaan, se on niin täydellisesti vierasta hänen
maailmankatsantokannalleen:
"Sinä, Mishka, olet siepannut päähäsi kirkollisia ajatuksia,
ikäänkuin olisit varastanut kurkkuja vieraasta yrttitarhasta ja
hämmennät tässä ihmisiä! Jos sanot, että työläiskansa on kutsuttu
uusimaan elämää, niin uusi se, äläkä kerää tomusta sitä, minkä papit
ovat pitäneet rikki ja heittäneet pois!"
Hänen kannaltaan se on oikein, hän on luokkatietoinen työmies,
hän elää luokassaan. Jos hän kuolee, niin se on hänestä
luokkaruumiin osan aineellisen muodon muuttamista, jotta koko
luokka voisi elää, jatkaa elämäänsä ja taisteluansa ja hän muassa
siinä; ikäänkuin ihminen leikkaa kyntensä pois ja kynsi putoaa
multaan, josta ihmisen ruumis saa ravintonsa elämänsä jatkamiseksi,
samaten kuolee työläinen, osa hänen luokkansa ruumiista, jotka hän
itse, hänen koko ruumiinsa, koko luokkansa voisi elää. Työläinen ei

kuole koskaan, koska sen luokka ei kuole. Hänen ja hänen luokkansa
edut ovat samat, vaistomaisesti ovat taloudelliset olot yhdistäneet
työläisen hänen luokkansa kanssa samaksi kokonaisuudeksi. Ja setä
Pjotr köyhälistöluokan edustajana ei kaipaa jumalia eikä turhia
kysymyksiä siitä, mistä ihminen tulee, mihin hän menee. Hän tietää
ne, ei järkiperäisesti, vaan vaistomaisesti, hänen mielestään nämä
kysymykset ovat järjettömiä.
Setä Pjotrin on ollut helppo saada sellainen
maailmankatsantokanta. Hän on työmies, hänen taloudellinen
asemansa on vaistomaisesti painanut hänen mieleensä sen, että
hänen ja toveriensa edut ja elämä ovat yhteiset, samanlaiset. Mutta
meidän ja Gorjkin jumalanhakija on kotosin pikkuporvarillisesta
talonpoikaiskylästä, jossa vielä kaikki "minä" vaistot kukostavat. Hän
on joutunut tuon kotipiirinsä ulkopuolelle, sentakia hän onkin
ruvennut epäilemään ja hakemaan, mutta hänen taloudellinen
asemansa ei selvitä hänelle suorastaan niitä kiusaavia elämän ja
kuoleman kysymyksiä. Hän hakee tukea niillä kysymyksillä, koska
hänellä ei ole tukea elämässä, koska hän on yksinäinen, sentakia
hän hakee Jumalaa, tiedustelee mistä hän on tullut ja mihin pitäisi
mennä.
Ja koska hän ei löydä enää vastausta vaistomaisesti, täytyy
hänelle sen selvitä järjen avulla.
Samassa asemassa ovat kautta maailman ne ihmiset, jotka jokin
silmiinpistävä taloudellinen etu on tuonut työväenpuolueeseen, vaan
joille kuitenkaan siitä huolimatta vielä ei vaistomaisesti tule
luokkatietoisuutta, eheätä yhteistunnetta koko työväenluokan
kanssa, sellaista tunnetta, että jokainen elää tuossa luokassa, ettei
kenellekään tule minkäänlaisia kysymyksiä kuoleman tahi elämän

suhteen ulkopuolella luokkaa. Mutta he kuitenkin pyrkivät yhtymään
työväenluokan maailmankatsantokantaan kaikkine vanhoine
käsitteineen, he hakevat jumalaa ja — kykeneekö sosialidemokratia
tarjoamaan sitä heille?
He hakevat jumalaa, s.t.s. he kysyvät: mihin joudumme
kuolemamme jälkeen? He hakevat tukea elämässä, koska heillä ei
ole sitä, mutta samalla myöskin tukea kuolemassa. Kykeneekö
köyhälistön maailmankatsantokanta antamaan heille sitä?
Opettaja Mihail tässä kirjassa koettaa auttaa jumalanhakijaamme,
osottaa sille, että jumalaa vielä ei ole olemassa, vaan että kansa
tulee luomaan ihmeitä tekevän jumalan, kansassa on pääpaino,
kansa on luonut kaiken maailmassa.
Mutta mikä jumala se on, jonka kansa tulee luomaan?
Ihmiskunta oli kerran eheä, silloin kun kaikkien ihmisten kokemus
oli samanlainen, silloin kuin kaikilla ihmisillä oli yhtäläiset
taloudelliset edut. Mutta ihmisten oli pakko taistelussa olemassaolon
edestä, ravinnon hankkimista varten luonnolta, antautua eri aloille;
spesialiseerautua taloudellisen elämän suhteen. Silloin ihmisen
kokemus tuli erilaisiksi, ihmisen "minä" erosi ihmiskunnan ja luonnon
kokonaisuudesta, yksityistalous toi ihmisille eri etuja, ihmiskunta
jakaantui orjiin ja herroihin, jakaantui kaikenlaisiin palasiin. Mutta
kerran tulee aika jolloin työväenluokka, yhdistäen etujen
samanlaisuuden kautta isomman osan ihmiskuntaa, aikaansaa taas
sellaisen järjestelmän, että kaikille ihmisille tulee sama kokemus ja
samat edut, että ihmiskunta tuntee taas itsensä eheäksi, jatkuvaksi,
kuolemattomaksi kokonaisuudeksi.

Gorjki lausuu opettaja Mihailin sanoilla: "Tulee aika jolloin koko
kansan tahto yhdistyy taas yhdeksi; silloin siinä herää
vastustamaton, ihmeellinen voima ja — silloin nousee jumala
kuolleista. Ja sitä te juuri haette…!"
Yhden sellaisista jumalanluomisen hetkistä näkee Gorjkin
jumalanluoja erään luostarin luona, jossa koko kansa, ja hän
mukana, yhdistää palavat toivomuksensa ja tahtonsa siihen, että
eräs raajarikko tyttö nousisi kävelemään; tyttö itse uskoo ja kansa
uskoo tahtonsa ja rukouksensa voimaan, ja tyttö tulee terveeksi.
Kansa tekee ihmeitä ja on aina maailmassa tehnyt ihmeitä, kansa
luo jumalan.
Mikä jumala se on?
Se on kaikkea auttava, kaikkea armahtava jumala, koska se on
kansan hyvä ja armelias tahto, joka rakastaa lapsiaan ja tekee hyvää
kaikille.
Jumalanhakijat, te jotka haette jumalaa tueksenne — katsokaa:
yhdistetyn kansan yhtenäinen tahto on voimakkain tuki mitä ihminen
voi ajatella. Säästää ja suojaahan ihminen jäseniään, miksei kansa
voisi samaten turvata jäseniään, joitten loukkaus tuntuu yhdistetyn
kansan ruumiissa!
Jumalanhakijat, te jotka pelkäätte kuolemaa! Jos tahtonne,
toimintanne, elämänne on sama kuin kansan elämä, niin olette
kuolemattomia, koska kansa on kuolematon. Jos voitte sanoa: kansa
ja minä olemme sama — kansan etu on minun etuni, kansan suru
minun suruni, silloin ei ole kuolemaa. Jos voitte sanoa, että taistelu

itsenne vuoksi on samassa taistelu koko kansan edestä, silloin elätte
ikuisesti kuten taistelunne jäljetkin elävät.
Jumalanhakijat, jos tarkotatte jumalallanne onnen kaipausta, niin
löydätte onnenne siitä yhdistetystä kansantahdosta. Muistatteko
miten lapsena rukoillessanne sulauduitte yhteen koko maailman
kanssa suureen hämärään kokonaisuuteen ja olitte onnellisia —
lähtekää nyt sinne missä tuhannet ajattelevat, vaativat samaa, niin
tunnette miten sulaudutte taas ihmisiin, maailmaan, Jumalaan,
tulette hyviksi, onnellisiksi — kuten lapset!
* * * * *
Uskonnon siinä muodossa kun se nyt on, sosialismi kerran tulee
hävittämään, mutta uskonnollinen tunnelma, tunnelma siitä, että
koko maailma ja ihmisyys on sama, onnen tunnelma jota ihmiset
tähän asti ovat koettaneet saavuttaa epätoivossaan keksien jumalia
itselleen, se tunnelma tulee jäämään ja vahvistumaan ja taloudelliset
olot, yhdistämällä ihmisiä samaan taloudelliseen, taisteluun, tulevat
edistämään tämän tunnelman kartuttamista elämässä, joka on
ihmisen suurin onni.
* * * * *
Gorjki on tunnustanut miten hän, jolle ei alkuaan taloudellinen
asema ole antanut köyhälistön maailmankatsantokantaa, löysi sen,
kulkien jumalaa, suurta voittamatonta jumalaa hakemassa, sellaista
jumalaa, jonka ihmiset olivat riistäneet palasiksi.
Jumalanhakija löysi jumalansa kansan tahdosta!
Jospa pian, pian kaikki sen löytäisivät sieltä!

Silloin tulisivat ihmiset hyviksi, koska heillä ei olisi enää syytä olla
pahoja…
Helsingissä 17 päivänä maaliskuuta 1909.
Hella Wuolijoki.
Sallikaa että kerron elämäni; tämä kertomus ei vie teiltä
paljoakaan aikaa ja teidän kumminkin täytyy se tietää.
Minä olen nokkonen, löytölapsi, laiton ihminen; en tiedä kuka
minut lienee synnyttänyt, minut löydettiin herra Lossevin maatilalla,
Sokolin kylässä, Krasnoglinin piirikunnassa. Joko äitini tahi joku muu
oli asettanut minut herrasväen puistoon, sen kappelin rappusille,
johon vanha rouva Losseva on haudattu; Danila Vjalov, puutarhuri,
löysi minut sieltä. Hän tuli varhain aamulla puistoon ja näki, että
rukoushuoneen oven edessä liikkui lapsi räsyihin verhottuna ja sen
ympärillä käyskeli harmaa kissa kuin vartijana.
Danilan luona minä elin neljänteen ikävuoteeni saakka, vaan
hänellä oli itsellään paljon lapsia eikä minun ruuastani kukaan huolta
pitänyt, söin mitä sain, ja kun en enään mistään löytänyt ruokaa niin
porasin, porasin ja sitte nukuin nälkäsenä.
Kun tulin neljä vuotta vanhaksi, niin otti lukkari Larion minut
luoksensa; hän oli yksinäinen ja hyvin kummallinen ihminen; hän otti
minut ikävänsä takia. Hän oli pieni, pyöreä kuin pallo, ja kasvonsakin
olivat pyöreät; hiukset punaset ja ääni kimeä kuin naisen ääni ja
sydämensäkin oli kuin naisen sydän, niin hellä kaikille. Hän piti
viinasta ja joi sitä paljon; raittiina hän oli vaitelias, silmät aina

puoleksi kiinni ja oli aina sen näköinen kuin olisi hän jonkun rikoksen
tehnyt, vaan juovuksissa lauloi hän äänekkäästi kirkkolauluja, piti
päänsä suorana ja hymyili kaikille.
Hän oli kaukana ihmisistä, eli köyhästi, antoi maaosuutensa papille
ja itse kävi talvella ja kesällä kaloja pyytämässä, ja huvin vuoksi
joskus lintujakin, jonka viimemainitun konstin hän opetti minullekin.
Hän piti linnuista, eivätkä ne peljänneet häntä; ihan liikuttavaa on
muistella miten joskus punatulkku juoksenteli hänen päätänsä pitkin
ja sekaantui hänen punasiin hiuksiinsa. Tahi joskus se istuutui
olkapäälle ja katseli hänen suuhunsa, kumartaen pikkuista, viisasta
päätänsä. Ja joskus panee Larion pitkäkseen penkille, ripottaa
hamppua hiuksiinsa ja partaansa ja kas, jopa lentävät keltasirkut,
tikliäiset, tiaiset, tilhit hänen luoksensa, myllertävät lukkarin
hiuksissa, ryömivät hänen poskiansa pitkin, nokkivat hänen korviaan,
istuskelevat hänen leuoilleen, vaan hän makaa ja nauraa hohottaa,
silmät puoleksi kiinni ja puheskelee ystävällisesti niiden kanssa. Minä
kadehdin häntä siitä, koska minua linnut pelkäsivät.
Hellä sielu oli Larion, ja kaikki eläimet käsittivät sen; ihmisistä sitä
en sano, en sano sitä tuomitakseni heitä, vaan koska tiedän ettei
vatsaa hellyydellä täytetä. Talvella hänellä oli vaikeanpuolista: puita
ei ollut eikä ollut rahaa ostaa niitä, raha oli tullut juoduksi; mökissä
oli kylmä kuin kellarissa, ainoastaan lintuset visertelivät ja laulelivat
ja me makasimme kylminä uunin päällä, päällämme kaikki mikä vaan
voi suojella pakkaselta ja kuuntelimme lintusten laulua… Larion
vihelsi heille — hän olikin siinä aika mestari, ja hän itsekin näytti
isolta käpylinnulta: nenä oli iso, nokankaltainen ja pää punanen.
Sanoipa hän joskus minulle:

— Kas, kuunteleppas Motjka — minut oli ristitty Matvjeiksi —
kuuleppas!
Hän asettuu selälleen, kädet pään alle, tirkistää silmillään ja alkaa
kimeällä äänellänsä hyräillä jotain sielumessua. Linnut vaikenevat,
kuuntelevat ja sitten alkavat itsekin laulaa kilpaa, vaan Larionin ääni
kuuluu kumminkin yli kaiken ja linnut oikein vimmastuvat, erittäinkin
keltasirkut ja tikliäiset tahi rastaat ja kottaraiset. Ja toisinaan hän
lauloi, kunnes silmistään vuosi kyyneleitä, jotka kastelivat hänen
poskensa ja hänen kasvonsa kyynelien kostuttamina tulivat vallan
harmaiksi.
Sellainen laulu joskus kerrassaan pelotti ja kerran minä sanoin
hänelle hiljakseen:
— Miksi sinä setä aina laulat kuolemasta? Hän lakkasi, katsoi
minuun ja lausui nauraen:
— Älä pelkää, tyhmyri! Ei se merkitse mitään että minä laulan
kuolemasta, sehän on vaan kaunista! Kaikkein kauniinta
jumalanpalveluksessa on sielumessu: siinä on hellyyttä, sääliä
ihmistä kohtaan. Meillä ei osata sääliä muita kuin kuolleita!
Nämä sanat minä muistan hyvin, kuten kaikki hänen puheensa,
vaan silloin minä tietysti en voinut niitä ymmärtää. Ainoastaan
vanhana, vasta viisaimpina vuosina käsitetään oikein kaikki lapsena
kuultu.
Myös muistan kysyneeni häneltä: miksi Jumala niin vähän auttaa
ihmisiä?

— Se ei ole Hänen asiansa! selitti Larion minulle. — Auta itse
itseäsi, sitä varten sinulle järkesikin on annettu! Jumala on vaan sitä
varten, ettei kuolema pelottaisi, vaan miten elät — se on sinun
asiasi!
Liian pian minä vaan unhotin nämä hänen sanansa, ja kun ne
uudelleen muistin — oli se jo liian myöhään, olin jo kärsinyt paljon
turhia suruja.
Merkillinen mies hän oli! Kaikki ihmiset, kun ovat ongella, eivät
huuda eivätkä puhu, jotteivät vaan pelottaisi kaloja, — vaan Larion
laulaa lakkaamatta, tahi kertoo minulle kaikenlaisia pyhäin taruja tahi
puhuu Jumalasta ja aina hän sai kaloja liiaksikin. Lintujakin
pyydystetään hyvin varovaisesti, vaan hän viheltää koko ajan,
ärsyttää niitä, puhuu niiden kanssa ja kaikesta huolimatta lintuja
tulee vaan verkkoihin. Ja niinpä mehiläistenkin kanssa — kun
siirretään niitä tahi tehdään niille jotain muuta, niin vanhat
mehiläishoitajat rukoilevat ennen, eikä se silloinkaan vielä heille
onnistu, vaan täytyy heidän kutsua avuksi lukkari, hän lyö mehiläisiä,
likistää niitä, haukkuu, — ja tekee kumminkin kaiken vallan
loistavasti. Hän ei pitänyt mehiläisistä: ne olivat tehneet hänen
tyttärensä sokeaksi. Pikku tyttö, hän oli vasta kolmenvuotias, oli
kiivennyt mehiläispesän luo, jossa mehiläinen oli pistänyt häntä
silmään. Pikkusilmä tuli kipeäksi ja sitten sokeaksi, sen perästä
myöskin toinen; sitten pikkutyttö kuoli päänkivun takia, ja äiti tuli
mielipuoleksi…
Hän teki kaikki vallan toisin kuin muut ihmiset, oli minulle hellä
kuin äiti; kylässä minua ei juuri kovinkaan suosittu: elämä oli
ahdasta ja minä olin kaikille vieras, liikanainen ihminen… Ehkäpä
soinkin laittomasti jonkun muille kuuluvan suupalasen!

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Number Theory Proceedings Of The Turku Symposium On Number Theory In Memory Of Kustaa Inkeri May 31june 4 1999 Reprint 2013 Matti Jutila Editor Tauno Metsnkyl Editor

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Number Theory Proceedings Of The Turku Symposium On Number Theory In Memory Of Kustaa Inkeri May 31june 4 1999 Reprint 2013 Matti Jutila Editor Tauno Metsnkyl Editor

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