Asymptotic Harmonic Behavior in the Prime Number Distribution


We consider on x > 0, where the sum is over all primes p. If Φ is bounded on x > 0, then the Riemann hypothesis is true or there are infinitely many zeros . The first 21 zeros give rise to asymptotic harmonic behavior in Φ(x) defined by the prime numbers up to one trillion.

Share and Cite:

van Putten, M. (2014) Asymptotic Harmonic Behavior in the Prime Number Distribution. Applied Mathematics, 5, 2547-2557. doi: 10.4236/am.2014.516244.

1. Introduction

The Riemann-zeta function is the analytic extension of


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,


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,

By the symmetry


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




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


on, where


with summation over all primes. In what follows, we put


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


In (9), is evidently unbounded in the limit as approaches zero whenever a finite number of zeros

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]


based on the Chebyshev functions

Figure 1.Shown are the trajectories of continuation in the complex plane by numerical integration of (2) with initial data indicated by small dots on. Continuation produces roots indicated by open circles, defined by finite endpoints of in the limit as approaches infinity. The roots produced by the choice of initial data are the first three on and −2 and −4 of the trivial roots.


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.


Our analysis begins with some known properties of in, e.g.,[4] -[9] .

Riemann obtained an analytic extension of by expressing in terms of,




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)


which gives a maximal analytic continuation of and shows a simple pole at with residue 1.

Riemann further introduced the symmetric form, satisfying

, whereby


using and. Along, is nonvanishing [10] -[13] , allowing


in terms of the digamma function


in the limit of large.

Lemma 2.1. In the limit of large, the logarithmic derivative of satisfies


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

becomes large. Hence,the are absolutely summable. Numerically, their sum is small,

based on a large number of known zeros.Lemma 2.3. In the limit of large, we have


Proof. By Lemma 2.1-2.2, we have


for large. Also[4] [14] [15]


on for some positive constants.,

3.An IntegralRepresentation of

Following the same steps leading to the Riemann integral for, we have


where absorbs the simple pole in at due to the simple pole in at, leaving

analytic at. Following a decomposition,


and substitution, appears as the Laplace transforms


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


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


where is analytic at. By (22), as approaches from the right, we have


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 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

Figure 2.The top window shows on and its leading order approximation. The asymptotic harmonic behavior is apparent in the residual difference (52) between the two, shown in the bottom two windows, including the period of 2.2496 in associated with the first zero.


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

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



and its Fourier transform


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)


where is a constant, so that




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


in the limit of large.

Proof. Integration over gives


where we choose to be between two consecutive values of. We have


In the limit as approaches infinity, approaches zero and becomes small by Lemma 2.2., whence


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


that is, by (22) and (29),


We thus consider


which ab initio is defined on by Euler’s identity with Fourier transform


Turning to the right hand side of (40), we consider the coefficients


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


whenever. Hence, for, whenever. It follows that


on.Lemma 5.2.For any, the Fourier transform of over satisfies


Proof. The Fourier integral can be obtained in a contour integration with closure over and the edges

for large. In the notation (42), it obtains a residue

at, since, whence


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),


with a remainder


Lemma 5.3. For, the Fourier transform


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


By Proposition 4.2 and Lemmas 5.1-5.2, we have


With, Theorem 1.1 now follows. ,

6.Numerical Illustration of Asymptotic Harmonic Behavior

The harmonic behavior emerges in


To search for higher harmonics associated with the zeros in

, we compare the spectrum of by taking a Fast Fourier Transform with respect to,


and compare the results with an analytic expression for the Fourier coefficients of the,


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.


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 ofdefined 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


According to (9) and our numerical calculation shown in Figure 3,the zeros explored to large k by

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.


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.


1When is an integer, is one-half the surface area of.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Keller, H.B. (1987) Numerical Methods in Bifurcation Problems. Springer Verlag/Tata Institute for Fundamental Research, Berlin.
[2] Hadamard, J. (1893) Etude sur les propriétés des fonctions entiéres et en particulier d’une fonction. Journal de Mathématiques Pures et Appliquées, 9, 171-216.
[3] von Mangoldt, H. (1985) Zu Riemann’s Abhandlung “Über die Anzahl der Priemzahlen unter einer gegebenen Grösse”. Journal für die Reine und Angewandte Mathematik, 114, 255-305.
[4] Titchmarsh, E.C. (1986) The Theory of the Riemann Zeta-Function. 2nd Edition, Oxford.
[5] Lehmer, D.H. (1988) The Sum of Like Powers of the Zeros of the Riemann Zeta Function. Mathematics of Computation, 50, 265-273.
[6] Dusart, P. (1999) Inégalités explicites pour Ψ(X), θ(X), π(X) et les nombres premiers. Comptes Rendus Mathematiques (Mathematical Reports) des l’Academie des Sciences, 21, 53-59.
[7] Keiper, J.B. (1992) Power Series Expansions of Riemann’s ζ Function. Mathematics of Computation, 58, 765-773.
[8] Ford, K. (2002) Zero-Free Regions for the Riemann Zeta Function. Number Theory for the Millenium, 2, 25-26.
[9] Borwein, P., Choi, S., Rooney, B. and Weirathmueller, A. (2006) The Riemann Hypothesis. Springer Verlag, Berlin.
[10] Littlewood, J.E. (1922) Researches in the Theory of the Riemann ζ-Function. Proceedings of the London Mathematical Society, Series 2, 20, 22-27.
[11] Littlewood, J.E. (1926) On the Riemann Zeta-Function. Proceedings of the London Mathematical Society, Series 2, 24, 175-201.
[12] Littlewood, J.E. (1928) Mathematical Notes (5): On the Function 1/ζ(1+ti). Proceedings of the London Mathematical Society, Series 2, 27, 349-357.
[13] Wintner, A. (1941) On the Asymptotic Behavior of the Riemann Zeta-Function on the Line . American Journal of Mathematics, 63, 575-580.
[14] Richert, H.E. (1967) Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1. Mathematische Annalen, 169, 97-101.
[15] Cheng, Y. (1999) An Explicit Upper Bound for the Riemann Zeta Function near the Line σ = 1. Rocky Mountain Journal of Mathematics, 29, 115-140.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.