Asymptotic harmonic behavior in the prime number distribution

We consider $\Phi(x)=x^{-\frac{1}{4}}\left[1-2\sqrt{x}\Sigma e^{-p^2\pi x}\ln p\right]$ on $x>0$, where the sum is over all primes $p$. If $\Phi$ is bounded on $x>0$, then the Riemann hypothesis is true or there are infinitely many zeros Re~$z_k>\frac{1}{2}$. The first 21 zeros give rise to asymptotic harmonic behavior in $\Phi(x)$ defined by the prime numbers up to one trillion.

A general approach to find zeros of as endpoints of continuation (Keller 1987). Here, we set out to analyze the z k of as endpoints of continuation along a trajectory z(λ), whose tangent z ′ (λ) satisfies in response to a choice of initial data z(0) = z 0 , is a zero of ξ(z). The trajectory z(λ) continues whenever τ = 0, i.e., whenever Θ(z) = − ζ ′ (z) ζ(z) exists and is finite. Continuation halts only at a zero of 1/Θ(z), corresponding to a zero of ζ(z), all of which are isolated. Continuation (2) can equivalently be formulated in terms of the prime numbers by ξ(z) = Σ ln(p)p −z (Re z > 1), representing the leading term the analytic extension of the sum Σξ(mz) = Σ ln(p) by virtue Euler's identity ζ(z) = Π 1 + p −z + p −2z + · · · = Π 1 − p −z −1 .
All known nontrivial zeros determined by numerical root finding satisfy Re z k = 1 2 to within numerical precision, the first three of which are z 1 = 1 2 ± 14.1347i, z 2 = 1 2 ± 21.0220i, z 3 = 1 2 ± 25.0109i. By the symmetry -Shown are the trajectories of continuation z(λ) in the complex plane z by numerical integration of (2) with initial data z 0 = 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 = 1 2 and -2 and -4 of the trivial roots.
Theorem 1.1. In the limit as x > 0 becomes small, we have the asymptotic behavior Corollary 1.2. If Φ is bounded, then the Riemann hypothesis is true or there are infinitely many zeros Re z k > 1 2 . A similar relation between the distribution of z k and the primes is given by (Hadamard 1893;von Mangoldt 1895) based on the Chebyshev functions where the sum is over all primes p and integers k. In Theorem 1.1, normalization of Φ(x) in Theorem 1.1 is by x 1 4 according to (8) and Z is absolutely convergent for all x > 0, whereas in (12) normalization is by √ u and the sum Σ u z k − 1 2 z k is not absolutely convergent. Similar to Corollary 1.2, it will be appreciated that the left hand side of (12) will be bounded in the limit of large u if the Riemann hypothesis is true.
In §2, we give some background on ζ(z). In §3 we introduce an integral representation of ξ(z) and derive one of its integral properties associated with the singularity of ζ(z) at z = 1. In §4, we apply Cauchy's integral formula to ζ(z) to derive a sum of residues associated by the z k . The proof Theorem 1.1 is given in §5 by a Fourier transform and asymptotic analysis of the expanded Euler's identity. In §6, we report on a direct evaluation of Φ(x) using the primes up to one trillion, to show the harmonic behavior as as encoded in Z by the first few zeros z k . We summarize our findings in §7.
Riemann provided an analytic extension of ξ(z) to the entire complex plane by expressing each term n −z in terms of Γ z 2 , where satisfies the asymptotic property θ 1 (x) ∼ 1 2 √ x as x approaches zero by the identity θ(x −1 ) = √ xθ(x) for the Jacobi function θ(x). 1 Evaluation of the integral (14) on Re z > 1 hereby reveals the meromorphic expression (e.g. Borwein et al. 2006) 1 When z = n is an integer, the pre-factor π n 2 Γ( n 2 ) will be recognized to be half the surface area S(n) of a unit n−sphere.
representing a maximal analytic continuation of ζ(z), and the presence of a simple pole at z = 1 with residue 1.
Riemann also introduced the symmetric form Q(z)ζ(z), (Littlewood 1922(Littlewood , 1924(Littlewood , 1927Wintner 1941), which allows consideration of its logarithmic derivative, in terms of the digamma function in the limit of large |z|.
Lemma 2.1. In the limit of large y, the logarithmic derivative of ζ(z) satisfies Proof. The result follows from (19) and (18). 2 Lemma 2.2. Along the line z = iy, we have the asymptotic expansion |γ(iy)| ∼ 2π y e − π 2 y in the limit of large y, whereby the γ k are absolutely summable.
Proof. Recall (10) and the asymptotic expansion ] with a branch cut along the negative real axis. In the limit of large y k , and the asymptotic behavior of the zeros z k , whereby y k ∼ 2πk ln k , and hence |γ k | ∼ e − π 2 k ln k ,where we used | arg z k | ∼ π 2 in the limit of large k. It follows that the γ k are absolutely summable. Numerically, their sum is rather small, Σ|γ k | = 3.5 × 10 −5 based on a large number of known zeros z k . 2 Lemma 2.3. In the limit of large y, we have Proof. By Lemma 2.1-2, we have asymptotically for large y. By (Richert 1967;Cheng 1999;Titchmarsh 1986;Broughan 2005) on y > δ for some positive constants c, δ. 2 These observations give rise to the following.
Proposition 2.4. The Fourier transform where P V refers to the Principle Value about y = 0, is well defined and F (λ) → 0 in the limit of large λ.
Proof. The result follows from Lemma 2.1-3 and the Riemann-Lebesque lemma. 2 3. An integral representation of ξ(z) We begin with an integral representation of ξ(z) in terms of φ(x) and Φ(x) given in (8), following steps very similar to those by Riemann leading to (16). To this end, we define g(z) = g 1 (z) + g 2 (z), for Re z > 1 and, respectively, all z.
Proof. Following steps similar to those leading to (16), we have ξ , that may be decomposed in integration over (0, 1] and [1, ∞), where g 1 (z) includes regularization about z = 1 made explicit in (26) by 1 z−1 , where the singularity at z = 1 in ξ can be seen to result from the Prime Number Theorem in the form of 2 √ xφ(x) −1 = o(1) on the basis of the asymptotic behavior of the function (Dusart 1999).
Lemma 3.2. The analytic extension of g 1 (z) extends to the real values z > 1 2 . Proof. In a neighborhood of 0 < z < 1, we may write where u 1 (z) is analytic about z > 0. With z = a + ib, the second term on the right hand side in the expanded Euler's identity (7) satisfies whereby it is bounded in Re z = a > 1 2 . Since the second term ζ(2z) in (7) is analytic in Re z = a > 1 2 , it follows that g(a) as defined in Proposition 3.1 is analytic on a > 1 2 . In view of the analytic and finite behavior of the right hand side in (7), the first and second term on the left hand side in (7) remain balanced as a approaches 1 2 from the right, giving where u 2 (a) is analytic at a = 1 2 . As a approaches 1 2 from the right, we have where u 3 (a) is analytic about a = 1 2 . 2 By Proposition 3.2, the behavior of the sign of Φ(x) in a neighborhood of x = 0 is relevant. If Φ(x) is of one sign in some neighborhood, then g 1 (z) has an analytic extension into Re z > 1 2 with no singularities, implying the absence of z k here. However, this requires information on the point wise behavior of Φ(x), which goes beyond the relatively weaker integrability property (31). To make a step in this direction, we next apply a linear transform to (7) to derive the asymptotic behavior of Φ(x) in terms of the distribution z k .

A sum of residues Z associated with the non-trivial zeros
We next define and its Fourier transform Lemma 4.1. The function h(z) has a simple pole at z = 1 with residue 1 and simple poles at each of the nontrivial zeros z k of ζ(z) with residue γ k .
Proof. As a consequence of (e.g. Borwein et al. 2006) where B is a constant, so that Here includes contributions from the logarithmic derivative of the factor to ζ(z) in (34), whose singularities are restricted to the trivial zeros of ζ(z). 2 Proposition 4.2. The Fourier transform of h(z) over Re z > sup a k satisfies in the limit of large λ < 0.
Proof. Consider contour integration over the rectangle Re z = 0, Re z = a and z = x±iY (0 < x < a). The line integral over z = 0 satisfies O e λ 2 by Proposition 2.4. Integration where we choose Y to be between two consecutive values of y k . We have In the limit as k approaches infinity, y k − Y approaches zero and |γ k | becomes small by Lemma 2.2., whence The result now follows in the limit as k approaches infinity, taking into account the residue sum e − λ 2 Z(λ) associated with the z k and absolute summability of the γ k . 2 5. Proof of Theorem 1.1 Following the expanded Euler identity (7), we define for all z away from the z k , its Fourier transform on a > sup a k , and the coefficients for all m ≥ 1. We remark that in particular, and m −1 c m (m −1 z) = 1 + 1 2 ln π + 1 2 γ z + O (z 2 ) has a well defined limit and C m → 2 in the limit as m becomes arbitrarily large.
Proof. Since r N (z) is analytic in Re z > 1 2 , its transform R N (λ) exists and is uniquely defined for a > 1 2 . The result follows from the Riemann-Lebesque lemma. 2 Proof of Theorem 1.1. The Fourier transform of the expanded Euler identity in the form (49) is By Propositions 5.1, 4.2 and Lemmas 5.1-5.4, we have With x = e 2λ , Theorem 1.1 now follows. 2 6. Numerical illustration using the primes up to one trillion Fig. 2 shows the first harmonic in the normalized density Φ(x) for small x evaluated for the 37.6 billion primes up to y = 10 12 , allowing x down to 2.6 × 10 −23 (λ = −26) in view of the requirement for an accurate truncation in φ(x) as defined by (8). The harmonic behavior is emergent in To search for higher harmonics Z i (λ) associated with the zeros z i in λǫ [−26, −11.7759] = [λ 1 − λ 2 , λ 1 + λ 2 ], we compare the spectrum of Φ e 2λ by taking a Fast Fourier Transform with respect to α, and compare the results with an analytic expression for the Fourier coefficients of the Z i (λ) (i = 1, 2, · · ·), where J n (z) denotes the Bessel function of the first of order n. Fig. 3 shows the first 21 harmonics in our evaluation of Φ(x), which appears to be about the maximum that can be calculated by direct summation in quad precision.

Conclusions
The zeros z k = a k + iy k of the Riemann-zeta function are endpoints of continuation: ξ(z k ) = 0 iff 1/Θ(z k ) = 0. By Euler's identity (5), continuation and hence the endpoints are controlled by an integral ξ(z) of a regularized sum Φ(e 2λ ) over the prime numbers as defined by (8).
The contributions of the zeros z k introduce asymptotic harmonic behavior in Φ e 2λ as a function of λ < 0 as defined by the sum Z(λ) of residues associated with the z k , shown in Figs. 1-2. This harmonic behavior in the prime number distribution appears to be exceptionally dense: 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 exponentially with the number of harmonics it contains. Conversely, knowledge of a relatively small number of zeros z k provides information on exponentially large primes numbers.
Theorem 1.1 introduces a correlation between Φ, defined over the primes, and Z, defined over the z k . Suppose there are a finite number of zeros z k in Re z > 1 2 . We may then consider k * for which a k * = max a k gives rise to dominant exponential growth in Z(λ) in the limit as λ < 0 becomes large. This observation leads to Corollary 1.2. Z can remain bounded in x > 0 only if the Riemann hypothesis is true, or if Z(λ) remains fortuitously bounded as an infinite sum over a k > 1 2 with no maximum in a < 1. .7756] and its leading order approximation 1.3616 + 1.5332e λ 6 . The asymptotic harmonic behavior is apparent in the residual difference (53) between the two, shown in the bottom two windows, including the period of 2.2496 in λ associated with the first zero z * = 1 2 ± 14.1347i.  (54), where λ 1 = −26, λ 2 = −11.7756] covers 32 periods of Z 1 (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 c ni [λ 1 , λ 2 ] of the Z i (λ) given by the analytic expression (55) for i = 1, 2, 3, · · · (continuous line). Shown are also the individual spectra of Z i (λ) for i = 1, 8 and 15 associated with the zeros z 1 , z 8 and z 15 . 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 z i of ζ(z).