1. Introduction
The Riemann-zeta function is the analytic extension of
(1)
where Euler’s identity on the right hand side expresses the relation of the integers to the primes. The zeros
of Riemann’s analytic continuation of (1) comprise the negative even integers,
, and an infinite number of nontrivial zeros
in the strip
.
A general approach to find zeros is by continuation[1] . If
is a starting point of a path
with tangent
,
(2)
then the endpoint
is a zero of
, all of which are isolated. All known nontrivial zeros satisfy
to within numerical precision, the first three of which are
,
![](https://www.scirp.org/html/htmlimages\12-7402176x\63bbdd31-e884-4636-9e46-d0599ded0887.png)
By the symmetry
(3)
it suffices to study zeros in the half plane
. Figure 1illustrates root finding by (2) for the first few zeros.
Continuation (2) is determined by the prime numbers, since
(4)
whereby
(5)
The poles of
at the zeros are therefore expressed by the prime number distribution.
In this paper, we study the distribution of zeros
by Fourier analysis of the function
(6)
on
, where
(7)
with summation over all primes. In what follows, we put
(8)
The
are absolutely summable by Stirling’s formula and the asymptotic distribution of
.
Theorem 1.1. In the limit as
becomes small, we have the asymptotic behavior
(9)
In (9),
is evidently unbounded in the limit as
approaches zero whenever a finite number of zeros ![](https://www.scirp.org/html/htmlimages\12-7402176x\28e9f86b-bb56-449c-b22c-afef3493d5f9.png)
exists off the critical line
.
Corollary 1.2. If
is bounded, then the Riemann hypothesis is true or there are infinitely many zeros
.
A similar relation between the distribution of
and the primes is [2] [3]
(10)
based on the Chebyshev functions
(11)
where the sum is over all primes
and integers
. In (9),
has a normalization by
according to and
is absolutely convergent for all
, whereas in (10)
is normalized by
and the sum
is not absolutely convergent. Similar to Corollary 1.2, the left hand side of (10) will be bounded in the limit of large
if the Riemann hypothesis is true.
Section 2 presents some preliminaries on
. Section 3 gives an integral representation of
and a discussion on its singularity at
. In Section 4, Cauchy’s integral formula is applied to derive a sum of residues associated with the
. The proof Theorem 1.1 follows from a Fourier transform and asymptotic analysis (Section 5). In Section 6, we illustrate a direct evaluation of
using the primes up to one trillion, showing harmonic behavior arising from
by the first few zeros
. We summarize our findings in Section 7.
2.Background
Our analysis begins with some known properties of
in, e.g.,[4] -[9] .
Riemann obtained an analytic extension of
by expressing
in terms of
,
(12)
where
(13)
Here,
satisfies
as
approaches zero by the identity
for the Jacobi function
1. On
, it obtains the meromorphic expression (e.g. Borwein et al., 2006)
(14)
which gives a maximal analytic continuation of
and shows a simple pole at
with residue 1.
Riemann further introduced the symmetric form
,
satisfying
, whereby
(15)
using
and
. Along
,
is nonvanishing [10] -[13] , allowing
(16)
in terms of the digamma function
(17)
in the limit of large
.
Lemma 2.1. In the limit of large
, the logarithmic derivative of
satisfies
(18)
Proof. The result follows from (17) and (16). Lemma 2.2.Along the line
, we have the asymptotic expansion
in the limit of large
, whereby the
are absolutely summable.
Proof.Recall (8) and the asymptotic expansion
with a branch cut along the negative real axis. In the limit of large
,
, and hence
, since
as ![](https://www.scirp.org/html/htmlimages\12-7402176x\5f602dcc-9819-4042-b2ed-66747f8e8581.png)
becomes large. Hence,the
are absolutely summable. Numerically, their sum is small,![](https://www.scirp.org/html/htmlimages\12-7402176x\673c02d0-3276-4602-9453-aaaa08b2a63b.png)
based on a large number of known zeros
.Lemma 2.3. In the limit of large
, we have
(19)
Proof. By Lemma 2.1-2.2, we have
(20)
for large
. Also[4] [14] [15]
(21)
on
for some positive constants
.,
3.An IntegralRepresentation of ![](https://www.scirp.org/html/htmlimages\12-7402176x\25457553-ccb8-4c0a-97f7-de946eea2dde.png)
Following the same steps leading to the Riemann integral for
, we have
(22)
where
absorbs the simple pole in
at
due to the simple pole in
at
, leaving
analytic at
. Following a decomposition
,
(23)
and substitution
,
appears as the Laplace transforms
(24)
These integral expressions allow continuations to
, respectively, the entire complex plane.
Lemma 3.1. Analytic extension of
extends to
.
Proof. With
, the second term on the right hand side in (5) satisfies
(25)
which is bounded in
. Since the second term
in (5) is analytic in
, it follows that
in is analytic on
. Following (5) as
approaches
from the right, we have
(26)
where
is analytic at
. By (22), as
approaches
from the right, we have
(27)
where
is analytic about
. Figure 2 shows a numerical evaluation of
for small
evaluated for the 37.6 billion primes up to one trillion, allowing
down to ![](https://www.scirp.org/html/htmlimages\12-7402176x\4db4f2ac-514b-44a4-9b23-3f1780f2755e.png)
in view of the requirement for an accurate truncation in
as defined by (7). The result shows asymptotic harmonic behavior in the limit as
becomes small.
If the integral
(28)
is absolutely convergent as
approaches zero, e.g., when
is of one sign in some neighborhood of
, as in the numerical evaluation shown in Figure 2, then
has an analytic extension into ![](https://www.scirp.org/html/htmlimages\12-7402176x\7c6d3850-c7ec-494d-bf61-d13550ef25bb.png)
with no singularities, implying the absence of
in this region. However, this requires information on the point wise behavior of
, which goes beyond the relatively weaker integrability property (23).
To make a step in this direction, we next apply a linear transform to (5) to derive the asymptotic behavior of
in terms of the distribution
.
4.A Sum of Residues ZAssociated with the Non-Trivial Zeros
Consider
(29)
and its Fourier transform
(30)
Lemma 4.1.
has a simple pole at
with residue 1 and simple poles at each of the nontrivial zeros
of
with residue
.
Proof. We have (e.g. Borwein et al. 2006)
(31)
where
is a constant, so that
(32)
Here
(33)
where
denotes the digamma function as before, includes contributions from the logarithmic derivative of the factor to
in (31), whose singularities are restricted to the trivial zeros of
. We now consider the Fourier integral over
as part of contour integration closed over
and
.
Proposition 4.2. The Fourier transform of
over
satisfies
(34)
in the limit of large
.
Proof. Integration over ![](https://www.scirp.org/html/htmlimages\12-7402176x\0307333a-b646-476d-81c2-18d8289b2ca1.png)
gives
(35)
where we choose
to be between two consecutive values of
. We have
(36)
In the limit as
approaches infinity,
approaches zero and
becomes small by Lemma 2.2., whence
(37)
Next, integration over
with a small semicircle around
obtains an
result in the limit of large
by application of Lemma 2.1-2.3 and the Riemann-Lebesgue Lemma. The result now follows in the limit as
approaches infinity, taking into account the residue sum
associated with the
and absolute summability of the
. ,
5.Proof of Theorem 1.1
Multiplying (5) by
, we have
(38)
that is, by (22) and (29),
(39)
We thus consider
(40)
which ab initio is defined on
by Euler’s identity with Fourier transform
(41)
Turning to the right hand side of (40), we consider the coefficients
(42)
Here,
since
. In particular,
and
has a well defined limit and
in the limit as
becomes arbitrarily large.
Lemma 5.1. The sum
is well-defined on
.
Proof. The result follows from the case
. By the Prime Number Theorem,
, whereby summation over the tails
satisfy
(43)
whenever
. Hence, for
,
whenever
. It follows that
(44)
on
.Lemma 5.2.For any
, the Fourier transform of
over
satisfies
(45)
Proof. The Fourier integral can be obtained in a contour integration with closure over
and the edges
![](https://www.scirp.org/html/htmlimages\12-7402176x\04950a5d-dae2-401b-8a0d-e8d1cc6dbcde.png)
for large
. In the notation (42), it obtains a residue ![](https://www.scirp.org/html/htmlimages\12-7402176x\b9c073ec-84f5-4bd9-8e08-4eed8eb66a63.png)
at
, since
, whence
(46)
The integral (46) exists by virtue of a removable singularity of
at
. It asymptotically decays to zero for large
when
by the Riemann-Lebesgue Lemma. We now consider (40) with (22),
(47)
with a remainder
(48)
Lemma 5.3. For
, the Fourier transform
(49)
in the limit of large
.
Proof. Since
is analytic in
, we are at liberty to consider the transform
on
. The result follows from the Riemann-Lebesgue Lemma. Proof of Theorem 1.1. The Fourier transform of (47) is
(50)
By Proposition 4.2 and Lemmas 5.1-5.2, we have
(51)
With
, Theorem 1.1 now follows. ,
6.Numerical Illustration of Asymptotic Harmonic Behavior
The harmonic behavior emerges in
(52)
To search for higher harmonics
associated with the zeros
in
, we compare the spectrum of
by taking a Fast Fourier Transform with respect to
,
(53)
and compare the results with an analytic expression for the Fourier coefficients of the![](https://www.scirp.org/html/htmlimages\12-7402176x\77c790f9-27ba-44a9-b518-347ae9259cd0.png)
,
(54)
where
denotes the Bessel function of the first of order
. Figure 3 shows the first 21 harmonics in our evaluation of
, which is about the maximum that can be calculated by direct summation in quad precision.
7.Conclusions
The zeros
of the Riemann-zeta function are endpoints of continuation, defined by an expressed by a regularized sum
over the prime numbers defined by (6).
The zeros
of
introduce asymptotic harmonic behavior in
as a function of
defined by the sum
of residues of the
, shown in Figure 2,Figure 3. Primes up to 4 billion are needed to identify the first 4 harmonics, up to 70 billion for the 10 and up to 1 trillion for the first 21. It appears that, effectively, the prime number range scales exponentially with the number of harmonics it contains.
Theorem 1.1 describes a correlation between the distribution of the primes and the distribution of the nontrivial zeros
. Suppose there are a finite number of zeros
in
. We may then consider
for which
gives rise to dominant exponential growth in
in the limit as
becomes large. This observation leads to Corollary 1.2.
can remain bounded in
only if the Riemann hypothesis is true, or if
remains fortuitously bounded as an infinite sum over
with no maximum in
.
Conversely,Riemann hypothesis implies
(55)
According to (9) and our numerical calculation shown in Figure 3,the zeros
explored to large k by
![](https://www.scirp.org/html/htmlimages\12-7402176x\62344c58-ec93-46c3-a00d-48376859d24f.png)
Figure 3.Shown are the absolute values of the Fourier coefficients
of
obtained by a Fast Fourier Transform (FFT) of (52) on the computational domain (53), where
,
covers 32 periods of
(dots), on the basis of the 37,607,912,2019 primes up to 1,000,000,000,0039. The resulting spectrum is compared with the exact spectra
of the
given by the analytic expression (54) for
(continuous line). Shown are also the individual spectra of
for
and 15 associated with the zeros
,
and
. The match between the computed and exact spectra accurately identifies the first 21 harmonics of
in
out of 22 shown, corresponding to the first 21 nontrivial zeros
of
.
existing numerical experiments effectively probe (and constrain) a distribution in primes which extends exponentially large in k.
Acknowledgements
The author gratefully acknowledges stimulating discussions with Fabian Ziltener and Anton F.P. van Putten. Some of the manuscript was prepared at the Korea Institute for Advanced Study, Dongdaemun-Gu, Seoul. This research was supported in part by the National Science Foundation through TeraGrid resources provided by Purdue University under grant number TG-DMS100033. We specifically acknowledge the assistance of VickiHalberstadt,RichRaymond and KimberlyDillman. The computations have been carried out using Lahey Fortran 95.
NOTES
1When
is an integer,
is one-half the surface area of
.