Asymptotic harmonic behavior in the prime number distribution

Asymptotic harmonic behavior in the prime number distribution

Maurice H.P.M. van Putten Yongsil-Gwan 614, Sejong University, 143-747 Seoul, South Korea
Abstract

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

1 Introduction

The Riemann-zeta function is the analytic extension of

 ζ(z)=1+12z+13z+⋯=Π(1−p−z)−1  (Rez>1), (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 (Keller, 1987). If is a starting point of a path with tangent ,

 τζ′(z)ζ(z)=−1, (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 , By the symmetry

 ζ(s)=ζ(1−s)χ(1−s)χ(s),  χ(s)=πs2Γ(s2),  Γ(z)=∫∞0tz−1e−tdt, (3)

it suffices to study zeros in the half plane Re(. Fig. 1 illustrates root finding by (2) for the first few zeros. Figure 1: Shown are the trajectories of continuation z(λ) in the complex plane z by numerical integration of (2) with initial data z0=1+ni (n=1,2,3,⋯) indicated by small dots on Re(z)=1. Continuation produces roots indicated by open circles, defined by finite endpoints of z(λ) in the limit as λ approaches infinity. The roots produced by the choice of initial data are the first three on Re z=12 and -2 and -4 of the trivial roots.

Continuation (2) is determined by the prime numbers, since

 −ζ′(z)ζ(z)=−∑lnppz−1=Σξ(mz),  ξ(z)=∑p−zlnp  (Rez>1), (4)

whereby

 ξ(z)=−ζ′(z)ζ(z)−∑m≥2ξ(mz). (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

 Φ(x)=x−14[1−2√xϕ(x)] (6)

on , where

 φ(x)=∑e−p2πxlogp (7)

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

 Z(λ)=∑αke−λ(zk−12),  αk=γ(zk),  γ(z)=Γ(z2)πz2. (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

 Φ(x)=12γ(12)+Z(ln√x)+13γ(13)x112+o(x112). (9)

In (9), is evidently unbounded in the limit as approaches zero whenever a finite number of zeros exists off the critical line Re .

Corollary 1.2. If is bounded, then the Riemann hypothesis is true or there are infinitely many zeros Re .

A similar relation between the distribution of and the primes is (Hadamard, 1893; von Mangoldt, 1895)

 u−ψC(u)√u=∑uzk−12zk+ln(2π)+ln√1−u−2√u (10)

based on the Chebyshev functions

 ψC(u)=∑pk≤uln(p),  ϑC(u)=∑p≤ulnp, (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.

§2 presents some preliminaries on . §3 gives an integral representation of and a discussion on its singularity at . In §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 (§5). In §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 §7.

2 Background

Our analysis begins with some known properties of in, e.g., Titchmarsh (1986); Lehmer (1988); Dusart (1999); Keiper (1992); Ford (2002)).

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

 γ(z)ζ(z)=∫∞0xz2−1θ1(x)dx, (12)

where

 θ1(x)=θ(x)−12,  θ(x)=∞∑n=−∞e−n2πx. (13)

Here, satisfies as approaches zero by the identity for the Jacobi function . 111When is an integer, is one-half the surface area of . On Re, it obtains the meromorphic expression (e.g. Borwein et al. 2006)

 γ(z)ζ(z)=1z(z−1)+f(z),  f(z)=∫∞1(xz2−1+x−z2−12)θ1(x)dx, (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

 ζ(z)=πz−1ζ(1−z)Γ(12−z2)Γ(z2)=πzζ(1−z)cos(12πz)Γ(z2)Γ(12+z2)=πz−122z−1ζ(1−z)cos(12πz)Γ(z) (15)

using and . Along , is non-vanishing (Littlewood, 1922, 1924, 1927; Wintner, 1941), allowing

 ζ′(z)ζ(z)=−ζ′(1−z)ζ(1−z)+ln(2π)+π2tan(πz2)−ψ(z) (16)

in terms of the digamma function

 ψ(z)=Γ′(z)Γ(z)∼ln(z)+O(z−1) (17)

in the limit of large .

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

 ζ′(iy)ζ(iy)=−ζ′(1−iy)ζ(1−iy)+O(lny). (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 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

 ∣∣∣γ(iy)ζ′(iy)ζ(iy)∣∣∣=O(y−12e−π2ylny). (19)

Proof. By Lemma 2.1-2, we have

 ∣∣∣γ(iy)ζ′(iy)ζ(iy)∣∣∣∼√2πy(∣∣∣ζ′(1−iy)ζ(1−iy)∣∣∣+O(lny))e−π2y (20)

for large . Also (Richert, 1967; Cheng, 1999; Titchmarsh, 1986; bra05)

 ∣∣∣ζ′(1−iy)ζ(1−iy)∣∣∣≤c(lny)23(lnlny)13 (21)

on for some positive constants .

3 An integral representation of ξ(z)

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

 γ(z)ξ(z)=∫∞0xz2−1φ(x)dx=1z−1+g(z), (22)

where absorbs the simple pole in at due to the simple pole in at , leaving analytic at . Following a decomposition ,

 g1(z)=12∫10x2z−14Φ(x)dxx,  g2(z)=∫∞1xz2−1φ(x)dx, (23)

and substitution , appears as the Laplace transforms

 g1(z)=∫0−∞Φ(e2λ)eλ(z−12)dλ,  g2(z)=2∫∞0φ(e2λ)eλzdλ. (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

 ∑m≥3|ξ(mz)|≤∑n≥3n−3alogn1−n−a<−√2ζ′(3a)√2−1, (25)

which is bounded in Re . Since the second term in (5) is analytic in Re , it follows that in is analytic on . Following (5) as approaches from the right, we have

 ξ(a)=−12a−1+u1(a), (26)

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

 g1(a)=−12a−1+u2(a), (27)

where is analytic about .

Fig. 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. Figure 2: The top window shows Φ(e2λ) on λϵ[−26,−11.7756] and its leading order approximation 1.3616+1.5332eλ6. 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 λ associated with the first zero z∗=12±14.1347i.

If the integral

 ∫1ϵx2z−14Φ(x)dxx (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 Fig. 2, then has an analytic extension into Re  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 Z associated with the non-trivial zeros

Consider

 h(z)=γ(z)ζ′(z)ζ(z)+1z−1 (29)

and its Fourier transform

 H(λ)=∫a+i∞a−i∞h(z)e−λzdz2πi. (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)

 ζ1(z)=12z(z−1)γ(z)ζ(z),  ζ′1(z)ζ1(z)=B+Σk(1z−zk+1zk), (31)

where is a constant, so that

 γ(z)ζ′(z)ζ(z)+1z−1=γ(z)[B+Σk(1z−zk+1zk)]+A(z). (32)

Here

 A(z)=1−γ(z)z−1−γ(z)z−2γ(z)ψ(z)lnπ, (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 Re  as part of contour integration closed over and Re .

Proposition 4.2. The Fourier transform of over Re  satisfies

 H(λ)=e−λ2Z(λ)+O(1) (34)

in the limit of large .

Proof. Integration over gives

 e−iY∫iY+aiYh(z)e−λxdx2πi=e−iY∫iY+aiYΓ(z2)πz2ζ′(z)ζ(z)dx2πi+O(Y−1), (35)

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

 ∫iY+aiYΓ(z2)πz2ζ′(z)ζ(z)dx2πi∼γk∫iY+aiYζ′(z)ζ(z)dx2πi∼γk2πiln(2a−1)[1+4ai(yk−Y)1−2a+πi]. (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 2.1-3 and the Riemann-Lebesque 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

 γ(z)ξ(z)=−γ(z)ζ′(z)ζ(z)−γ(z)∑m≥2ξ(mz), (38)

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

 1z−1+g(z)=−h(z)+1z−1−γ(z)∑m≥2ξ(mz). (39)

We thus consider

 g1(z)=g2(z)+h(z)+γ(z)∑m≥2ξ(mz), (40)

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

 G1(λ)=∫a+i∞a−i∞g1(z)e−λzdz2πi=e−λ2Φ(e2λ). (41)

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

 cm(z)=γ(z)γ(mz),  Cm=1mγ(1m)  (m≥1). (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 Re .

Proof. The result follows from the case . By the Prime Number Theorem, , whereby summation over the tails satisfy

 ∞∑k≥nln(pk)p2ak∼∞∑k≥n[1k2aln(k)2a−1+lnln(k)(kln(k))2a]<∞ (43)

whenever . Hence, for , whenever . It follows that

 |Σm≥2ξ(mz)|≤Σm≥2Σpp−(m−2)ap−2alnp≤Σm≥02−mΣpp−2alnp=Σpp−2alnp<∞ (44)

on Re .

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

 Dm(λ)=Cme−λm+o(1) (45)

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

 Dm(λ)=Cme−λm+eλ2∫∞−∞cm(iy)imy−1e−iλydy2π. (46)

The integral (46) exists by virtue of a removable singularity of at . It asymptotically decays to zero for large when by the Riemann-Lebesque Lemma.

We now consider (40) with (22),

 g1(z)=g2(z)+h(z)+∑m≥2(cm(z)mz−1+cm(z)g(mz))=h(z)+N∑m=2cm(z)mz−1+rN(z) (47)

with a remainder

 rN(z)=g2(z)+∑m≥2cm(z)g(mz)+γ(z)∑m≥N+1ξ(mz). (48)

Lemma 5.3. For , the Fourier transform

 eλ2RN(λ)=∫a+i∞a−i∞rN(z)e−λ(z−12)dz2πi=o(1) (49)

in the limit of large .

Proof. Since is analytic in Re , we are at liberty to consider the transform on . The result follows from the Riemann-Lebesque Lemma.

Proof of Theorem 1.1. The Fourier transform of (47) is

 G1(λ)=H(λ)+D2(λ)+D3(λ)+R3(λ). (50)

By Proposition 4.2 and Lemmas 5.1-5.2, we have

 e−λ2Φ(e2λ)=e−λ2Z(λ)+C2e−λ2+C3e−λ3+o(e−λ2). (51)

With , Theorem 1.1 now follows.

6 Numerical illustration of asymptotic harmonic behavior

The harmonic behavior emerges in

 R(x)=−Φ(x)−C2−C3x112. (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 ,

 λ(α)=λ1+λ2cosα  (αϵ[0,2π]), (53)

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

 cni[λ1,λ2]=2Re{(−i)nγie−iλ1ziJn(−λ2zi)}, (54)

where denotes the Bessel function of the first of order . Fig. 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. Figure 3: Shown are the absolute values of the Fourier coefficients cn[λ1,λ2] of Φ(e2λ) obtained by a Fast Fourier Transform (FFT) of (52) on the computational domain (53), where λ1=−26, λ2=−11.7756] covers 32 periods of Z1(λ) (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 cni[λ1,λ2] of the Zi(λ) given by the analytic expression (54) for i=1,2,3,⋯ (continuous line). Shown are also the individual spectra of Zi(λ) for i=1,8 and 15 associated with the zeros z1, z8 and z15. The match between the computed and exact spectra accurately identifies the first 21 harmonics of Z(λ) in Φ out of 22 shown, corresponding to the first 21 nontrivial zeros zi of ζ(z).

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 Figs. 2-3. 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 the prime number range scales approximately 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 Re . 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

 limx→0+Φ(x)=12γ(12)≃1.3616. (55)

According to (9) and our numerical calculation shown in Fig. 3, the number of primes relevant to the observed asymptotic harmonic behavior scales approximately exponentially with the number of zeros . The zeros explored to large values by existing numerical experiments hereby constrain the distribution of an exponentially large number of primes.

Acknowledgment. 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 Vicki Halberstadt, Rich Raymond and Kimberly Dillman. The computations have been carried out using Lahey Fortran 95.

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