Zeta Zero-Counting Function

In this presentation, a more accurate expression of the zeta zero-counting function is developed exhibiting the expected step function behavior and its relation to the primes is demonstrated.

1. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Zeta Zero-Counting Function
Prof. Fayez A. Alhargan, PhD. BEng.
21 July 2021

2. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Background

3. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Introduction
The estimation of the number of zeta zeros was introduced by
Riemann in his landmark paper in 1859, as an approximate
expression. Then in 1905 H. von Mangoldt proved the
approximation. Also, Aleksandar Ivic restated the proof using
contour integration.
In this presentation, a more accurate expression of the zeta
zero-counting function is developed exhibiting the expected step
function behavior and its relation to the primes is demonstrated.
This presentation is based on the paper:
Alhargan, Fayez (2021),A Concise Proof of the Riemann Hypothesis via Hadamard
Product,https://hal.archives-ouvertes.fr/hal-03294415/document.

4. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Riemann Expression
In the rangef0; Tg, the number of roots of(s); was conjectured by
Riemann [1], as approximately
=
T
2
ln
T
2

T
2
; (1)
and some 46 years later was proved by H. von Mangoldt [2], the prove was
outlined by Ivic ([3], p. 17), where he showed using contour integration, that
the number of zeros is given approximately by
N(T) =
T
2
ln
T
2

T
2
+
7
8
+
1

=
Z
L

0
(s)
(s)
ds; (2)
and demonstrated that
=
Z
L
h

0
(s)
(s)
i
ds=O(lnT): (3)
Although the integral in Equation (3) is small compared to the major elements
in Equation (2), it still contains the sawtooth-like waveform component, that I
will demonstrate later.

5. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Now, recalling Riemann [1] main justication of Equation (1), quoted as
follows:
"because the integral
R
d log(t), taken in a positive sense around the
region consisting of the values oftwhose imaginary parts lie between
1
2
i
and
1
2
iand whose real parts lie between0andT, is (up to a fraction
of the order of magnitude of the quantity
1
T
) equal to(Tlog
T
2

T)i; this integral however is equal to the number of roots of(t) =
0lying within this region, multiplied by2i. One now nds indeed
approximately this number of real roots within these limits, and it is
very probable that all roots are real."
In essence, Riemann instinctively was invoking,
for(s)is a meromorphic function inside and on some closed contourD, and
(s)has no zeros or poles onD, thus
1
2i
I
D

0
(s)
(s)
ds=ZP; (4)
whereZandPdenote the number of zeros and poles of(s); inside the
contourD.

6. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary New Formulation

7. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Formal Denition
Figure (1) shows that the zero-counting function is a staircase step function.
Thus, we can utilize the Heaviside function, to formally dene the
zero-counting function(t), as
(t) :=
X
m
H(ttm);
(5)
where the sum is overtm, the imaginary values of the(s)zeros.
(t)t14.1321.0225.0130.421234
Figure 1:

8. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Cauchy's Argument Principle
Now, from the proof of the Riemann Hypothesis [4], which implies that(s)
has simple zeros only on the critical line<(s) =
1
2
, ats=smands= sm.
Then, we can invoke, to dene the
zero-counting function(t), in the rangefs;sgenclosed by the contourD, see
Figure (2), for the number of zeros of(s), as
4i(t) =
I
D
h

0
(s)
(s)
i
ds: (6)
Noting that(s) =
1
2
(s)(s1)s(
s
2
)

s
2, taking the log and dierentiating,
Equation (6) can be expressed in terms of zeta function as
4i(t) =
I
D


0
(s)
(s)
+
1
(s1)
+
1
s
+

0
(
s
2
)
(
s
2
)

1
2
ln

ds; (7)
where the closed contourDencompasses the critical strip[0 <(s)1].

9. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
DL1L2t
1
2
smssms
Figure 2:(s)Critical Strip, ContoursD,L1andL2.
From Figure (2), we see that
all the poles of[
0
(s)=(s) +
0
(
s
2
)=(
s
2
)]are on the critical line<(s) =
1
2
,
the contourL1encloses all thesmpoles in the range fromstos,
the contourL2encloses only the two poless= 0ands= 1.

10. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Thus, we can rearrange Equation (7), as
4i(t) =
I
L1


0
(s)
(s)
+

0
(
s
2
)
(
s
2
)

1
2
ln

ds+
I
L2
h
1
(s1)
+
1
s
i
ds:(8)
Now, the contour integration aroundL1, can be transformed to a
integral, as
I
L1
= lim
!0
Z
s+
s+
+
Z
s
s+

Z
s
s

Z
s+
s

= 2
Z
s
s
(9)
and using L2, we obtain
I
L2
h
1
(s1)
+
1
s
i
ds= 4i; (10)
thus, Equation (8) becomes
2i(t) =
Z
s
s


0
(s)
(s)
+

0
(
s
2
)
(
s
2
)

1
2
ln

ds+ 2i: (11)

11. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Integrating Equation (11), we have
2i(t) =

ln(s) + ln (
s
2
)
s
2
ln

s
s
+ 2i: (12)
Thus, nally we have
2i(t) = ln(s)ln(s) + ln (
s
2
)ln (
s
2
) + ln

s
2ln

s
2+ 2i;
(13)
withs=
1
2
+it, we have
2i(t) = ln(
1
2
+it)ln(
1
2
it)
+ ln (
1
4
+i
t
2
)ln (
1
4
i
t
2
)itln+ 2i:
(14)
Dierentiating Equation (14), we have
2i
0
(t) =

0
(
1
2
+it)
(
1
2
+it)


0
(
1
2
it)
(
1
2
it)
+

0
(
1
4
+i
t
2
)
(
1
4
+i
t
2
)


0
(
1
4
i
t
2
)
(
1
4
i
t
2
)
iln:
(15)

12. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Approximation

13. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Now, utilizing the rst few terms in the Stirling approximation; i.e.
(z)(z)
z
e
z
z

1
2
p
2; (16)
and the rst few terms in the asymptotic expansion of the digamma function;
i.e.

0
(z)
(z)
lnz
1
2z
: (17)
Therefore, we have
ln (
s
2
)ln (
s
2
)
s
2
ln
s
2

s
2
ln
s
2
+
1
2
ln s
1
2
lnsit; (18)
and

0
(
1
4
+i
t
2
)
(
1
4
+i
t
2
)


0
(
1
4
i
t
2
)
(
1
4
i
t
2
)
= ln(
1
2
+it)
1
(
1
2
+it)
ln(
1
2
it)+
1
(
1
2
it)
:(19)

14. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Thus, nally we have
2i(t) =(
1
4
+i
t
2
) ln(
1
4
+i
t
2
)(
1
4
i
t
2
) ln(
1
4
i
t
2
)
+
1
2
ln(
1
4
i
t
2
)
1
2
ln(
1
4
+i
t
2
)
+ ln(
1
2
+it)ln(
1
2
it)itlne= 2i
X
m
H(ttm):
(20)
Also, from Equation (15), we have
2i
0
(t) = ln(
1
2
+it)
1
(
1
2
+it)
ln(
1
2
it) +
1
(
1
2
it)
+

0
(
1
2
+it)
(
1
2
+it)


0
(
1
2
it)
(
1
2
it)
iln= 2i
X
m
(ttm):
(21)

15. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Equation (20) is a very accurate approximation, and it is a sum of dierences
between complex numbers and their conjugates, thus the result will always be
imaginary number as expected.
Further approximation of the log part gives
(t) =
t
2
ln
t
2e
+
7
8
+
1
2i

ln(
1
2
+it)ln(
1
2
it)

: (22)
We note from Alhargan [4], that
ln(s) =
X
k2N
X
p
1
k
e
kslnp
=s
X
k2N
(ks); (23)
where(s)is thes-domain prime-counting function, given by the Laplace
transform of thex-domain prime-counting function(x), as
(s) =L f(x)g=
X
p
L fH(lnxlnp)g=
X
p
e
slnp
s
: (24)

16. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Now, from equation (23), we have
ln(s)ln(s) =
X
k
1
k
s(ks)
1
k
s(ks); (25)
i.e.
ln(
1
2
+it)ln(
1
2
it) = 2i
X
p
X
k
1
k
p

k
2sin(tklnp); (26)
giving
(t) =
t
2
ln
t
2e
+
7
8
+
1

X
p
X
k
1
k
p

k
2sin(tklnp): (27)
We observe from Equation (25), the direct relation between the
zero-counting function(t)and thes-domain prime-counting function(s).
Furthermore, we observe in Equation (27), the direct relationship between
the zeta zero-counting function(t), and.
Although the equation is slow for computational purposes, it reveals the
underlying relationship between the zero-counting function and the primes.
Moreover, it exposes the source of the sawtooth-like waveform eect as the
spectrum sum of the prime harmonics.

17. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Computation

18. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
The [ln(
1
2
+it)ln(
1
2
it)]of(t)
is shown in Figure (3). It is observed that the component magnitude is less
than one.
However, it has a vital contribution to the accuracy of the zero-counting
function; which turns it into a Heaviside staircase step function, as shown in
Figure (4), this vital component has been overlooked in the literature.
Figure 3: (t).

19. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Finally, Figure (4) shows comparisons between(t)andN(t), and it conrms
that the zero counting function is a Heaviside staircase step function, and its
dierential
0
(t)is an impulse Dirac delta function; as can be seen in Figure (5).
Figure 4:(t)vs.N(t).

20. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Figure 5:j
0
(t)j.

21. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary References

22. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
References
Riemann, Bernhard (1859)."Über die Anzahl der Primzahlen unter einer
gegebenen Grösse". Monatsberichte der Berliner Akademie.. In
Gesammelte Werke, Teubner, Leipzig(1892), Reprinted by Dover, New
York (1953).
Mangoldt, H. von (1905),Zur Verteilung der Nullstellen der
Riemannschen Funktion(t). Mathematische Annalen 1905, Vol. 60, pp.
1-19,https://gdz.sub.uni-goettingen.de/id/PPN235181684_0060
Ivic, A. (1985).The Riemann Zeta Function, John Wiley & Sons.
Alhargan, Fayez (2021),A Concise Proof of the Riemann Hypothesis via
Hadamard Product,
https://hal.archives-ouvertes.fr/hal-03294415/document.

23. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
Summary

24. Zeta Zero-Counting
Function
Fayez A. Alhargan
Background
New Formulation
Approximation
Computation
References
Summary
BackgroundNew FormulationApproximationComputationReferencesSummary
The Zero-Counting Function Proof on One Slide
The zero-counting function(t)of the number of zeros of(s), in the rangefs;sg
enclosed by the contourD, is dened as
4i(t) =
I
D
h

0
(s)
(s)
i
ds=
I
D
h

0
(s)
(s)
+
1
(s1)
+
1
s
+

0
(
s
2
)
(
s
2
)

1
2
ln
i
ds:
(28)
where the closed contourDencompasses the critical strip[0 <(s)1]. Thus,
4i(t) =
I
L
1
h

0
(s)
(s)
+

0
(
s
2
)
(
s
2
)

1
2
ln
i
ds+
I
L
2
h
1
(s1)
+
1
s
i
ds:(29)
Integrating, using line integral forL1and residue theorem forL2, we have
2i(t) = ln(s)ln(s) + ln (
s
2
)ln (
s
2
) + ln

s
2ln

s
2+ 2i:(30)
Utilizing Stirling approximation, we have
2i(t) =(
1
4
+i
t
2
) ln(
1
4
+i
t
2
)(
1
4
i
t
2
) ln(
1
4
i
t
2
)
+
1
2
ln(
1
4
i
t
2
)
1
2
ln(
1
4
+i
t
2
)
+ ln(
1
2
+it)ln(
1
2
it)itlne= 2i
X
m
H(ttm):
(31)
Q.E.D.
Fayez A. Alhargan