Zeta Zeros Research Articles

On the zeros of Epstein zeta functions near the critical line

Let Q be a positive definite quadratic form with integral coefficients and its discriminant D<0 and let E(s,Q) be the Epstein zeta function associated with Q. Assume that the class number of the quadratic imaginary field Q(D) is bigger than 1. Then we estimate the number of zeros of E(s,Q) in the region ℜs>σT(θ):=1/2+(log⁡T)−θ and T<ℑs<2T, to provide its asymptotic formula for fixed 0<θ<1 conditionally. Moreover, it is unconditional if the class number of Q is 2 or 3 and 0<θ<1/13.

HIGHER CORRELATIONS AND THE ALTERNATIVE HYPOTHESIS

AbstractThe Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

On the Generalized Benford law

We provide some properties of the Generalized Benford law – a flexible model for the distribution of significant digits – which accurately describes the pattern of leading digits in the sequences of prime numbers and of non-trivial Riemann zeta zeros.

More than five-twelfths of the zeros of zeta are on the critical line

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form (mu star Lambda _1^{star k_1} star Lambda _2^{star k_2} star cdots star Lambda _d^{star k_d}) is computed unconditionally by means of the autocorrelation of ratios of zeta techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of ζ(s)+λ1ζ′(s)logT+λ2ζ′′(s)log2T+⋯+λdζ(d)(s)logdT,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\zeta (s) + \\lambda _1 \\frac{\\zeta '(s)}{\\log T} + \\lambda _2 \\frac{\\zeta ''(s)}{\\log ^2 T} + \\cdots + \\lambda _d \\frac{\\zeta ^{(d)}(s)}{\\log ^d T}, \\end{aligned}$$\\end{document}where zeta ^{(k)} stands for the kth derivative of the Riemann zeta-function and {lambda _k}_{k=1}^d are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

Toward the Unification of Physics and Number Theory

This paper introduces the notion of simplex-integers and shows how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. A geometric analogue to the primality test is introduced: when [Formula: see text] is prime, it divides [Formula: see text] for all [Formula: see text]. The geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound [Formula: see text]-simplex or associated [Formula: see text] lattice is discovered, a deep understanding of the error factor of the prime number theorem would be realized — the error factor corresponding to the distribution of the non-trivial zeta zeros, which might be the mysterious link between physics and the Riemann hypothesis [D. Schumayer and D. A. W. Hutchinson, Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83 (2011) 307]. It suggests how quantum gravity and particle physicists might benefit from a simplex-integer-based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax.

Zeros of the Selberg zeta function for symmetric infinite area hyperbolic surfaces

In the present paper we give a simple mathematical foundation for describing the zeros of the Selberg zeta functions Z_X for certain very symmetric infinite area surfaces X. For definiteness, we consider the case of three funneled surfaces. We show that the zeta function is a complex almost periodic function which can be approximated by complex trigonometric polynomials on large domains (in Theorem 4.2). As our main application, we provide an explanation of the striking empirical results of Borthwick (Exp Math 23(1):25–45, 2014) (in Theorem 1.5) in terms of convergence of the affinely scaled zero sets to standard curves {mathscr {C}}.

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces$\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$with cusps and all finite-dimensional unitary representations$\unicode[STIX]{x1D712}$of$\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for$(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if$\unicode[STIX]{x1D6E4}$is cofinite and$\unicode[STIX]{x1D712}$is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for$\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface$\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$of finite or infinite area and any finite-dimensional unitary representation$\unicode[STIX]{x1D712}$of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

On the Riemann Hypothesis, complex scalings and logarithmic time reversal

Abstract An approach to solve the Riemann Hypothesis is revisited within the framework of the special properties of Θ (theta) functions, and the notion of C T invariance. The conjugation operation C amounts to complex scaling transformations, and the T operation t → ( 1 ∕ t ) amounts to the reversal l o g ( t ) → − l o g ( t ) . A judicious scaling-like operator is constructed whose spectrum E s = s ( 1 − s ) is real-valued, leading to s = 1 2 + i ρ , and/or s = real. These values are the location of the non-trivial and trivial zeta zeros, respectively. A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions, reveals that n o zeros exist off the critical line. The role of the C , T transformations, and the properties of the Mellin transform of Θ functions were essential in our construction.

The Fourier Transform of the Non-Trivial Zeros of the Zeta Function

The non-trivial zeros of the Riemann zeta function and the prime numbers can be plotted by a modified von Mangoldt function. The series of non-trivial zeta zeros and prime numbers can be given explicitly by superposition of harmonic waves. The Fourier transform of the modified von Mangoldt functions shows interesting connections between the series. The Hilbert-Pólya conjecture predicts that the Riemann hypothesis is true, because the zeros of the zeta function correspond to eigen values of a positive operator and this idea encouraged to investigate the eigenvalues itself in a series. The Fourier transform computations is verifying the Riemann hypothesis and give evidence for additional conjecture that those zeros and prime numbers arranged in series that lie in the critical, 1/2. positive upper half plane and over the positive integers, respectively.

The distribution of zeros of ζ′(s) and gaps between zeros of ζ(s)

Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ(s) and its derivative ζ′(s). As an application, we prove a stronger form of half of a conjecture of Radziwiłł [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ′(s) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22]. Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ′(s).

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt counting formula (without any truncations) for the number [Formula: see text] of zeroes in the critical strip with imaginary part greater than [Formula: see text] and less than or equal to [Formula: see text].

Some remarks on the differences between ordinates of consecutive zeta zeros

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le T}{(\gamma_{n+1}-\gamma_n)}^k \qquad(k>0) $$ are proved.

Isolating some non-trivial zeros of zeta

Isolating some non-trivial zeros of zeta

Distribution of zeta zeros and the oscillation of the error term of the prime number theorem

An 84-year-old classical result of Ingham states that a rather general zero-free region of the Riemann zeta function implies an upper bound for the absolute value of the remainder term of the prime number theorem. In 1950 Tur´an proved a partial conversion of the mentioned theorem of Ingham. Later the author proved sharper forms of both Ingham’s theorem and its conversion by Tur´an. The present work shows a very general theorem which describes the average and the maximal order of the error terms by a relatively simple function of the distribution of the zeta zeros. It is proved that the maximal term in the explicit formula of the remainder term coincides with high accuracy with the average and maximal order of the error term.

Simultaneous distribution of the fractional parts of Riemann zeta zeros

We investigate the simultaneous distribution of the fractional parts of $\{\alpha_1 \gamma, \alpha_2\gamma, \cdots, \alpha_n\gamma\}$, where $n\geq 2$, $\alpha_1$, $\alpha_2$, $\ldots$, $\alpha_n$ are fixed, distinct positive real numbers and $\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta-function.

Why the Glove of Mathematics Fits the Hand of the Natural Sciences So Well :How Far Down the (Fibonacci) Rabbit Hole Goes

Why does the glove of mathematics fit the hand of the natural sciences so well? Is there a good reason for the good fit? Does it have anything to do with the mystery number of physics or the Fibonacci sequence and the golden proportion? Is there a connection between this mystery (golden) number and Leibniz’s general question, why is there something (one) rather than nothing (zero)? The acclaimed mathematician G.H. Hardy (1877-1947) once observed: “In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.” Is this also true of great physics? If so, is there a simple “preestablished harmony” or linchpin between their respective ultimate foundations? The philosopher-mathematician, Gottfried Leibniz, who coined this phrase, believed that he had found that common foundation in calculus, a methodology he independently discovered along with Isaac Newton. But what is the source of the harmonic series of the natural log that is the basis of calculus and also Bernhard Riemann’s harmonic zeta function for prime numbers? On the occasion of the three-hundredth anniversary of Leibniz’s death and the one hundredth-fiftieth anniversary of the death of Bernhard Riemann, this essay is a tribute to Leibniz’s quest and questions in view of subsequent discoveries in mathematics and physics. (In the Journal of Interdisciplinary Mathematics, Dec. 2008 and Oct. 2010, I have already sympathetically discussed in detail Riemann’s hypothesis and the zeta function in relation to primes and the zeta zeros. Both papers were republished online in 2013 by Taylor and Francis Scientific Publishers Group.)

Can one hear the Riemann zeros in black hole ringing?

We elaborate on an entry of the AdS/CFT dictionary relating functional determinants: the determinant of the one-loop contribution to the effective gravitational action by bulk scalars in an asymptotically locally AdS background X, and the determinant of the two-point function of the dual operator (a.k.a. scattering matrix) at the conformal boundary M. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk gravitational partition function to a subleading large-N contribution in the boundary CFT partition function:The formula applies to quotients of AdS as well [1]. In the particular case of the BTZ black hole, a closed expression can be worked out in terms of an associated Patterson-Selberg zeta function ZBTZ(λ) [2]. The determinants can then be thought of as regularized products of either zeta zeros, scattering resonances or quasinormal frequencies [3]. In this sense, one could say that the zeros of ZBTZ(λ) can be heard in the spectrum of quasinormal modes of the BTZ black hole. The question we want to pose is whether a similar situation might exist for the celebrated zeros of the Riemann zeta function.

Statistics for ordinary Artin-Schreier covers and other $p$-rank strata

We study the distribution of the number of points and of the zeroes of the zeta function in different p-rank strata of Artin-Schreier covers over Fq when q is fixed and the genus goes to infinity. The p-rank strata considered include the ordinary family, the whole family, and the family of covers with p-rank equal to p − 1. While the zeta zeroes always approach the standard Gaussian distribution, the number of points over Fq has a distribution that varies with the specific family.

A note on the zeros of zeta and L-functions

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound for a large class of $L$-functions including those which arise from cuspidal automorphic representations of GL($m$) over a number field. We also prove a number of related results including bounding the order of vanishing of an $L$-function at the central point and bounding the height of the lowest zero of an $L$-function.

On the distribution (mod 1) of the normalized zeros of the Riemann zeta-function

We consider the problem whether the ordinates of the non-trivial zeros of ζ(s) are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros (xn) are uniformly distributed modulo 1. Applying the Piatetski-Shapiro 11/12 Theorem we show that, for 0<κ<6/5, the mean value 1N∑n≤Nexp⁡(2πiκxn) tends to zero. In the case κ=1 the Prime Number Theorem is sufficient to prove that the mean value is 0, but the rate of convergence is slower than for other values of κ. Also the case κ=1 seems to contradict the behavior of the first two million zeros of ζ(s). We make an effort not to use the RH. So our theorems are absolute. Let ρ=12+iα run through the complex zeros of zeta. We do not assume the RH so that α may be complex. For 0<κ<65 we prove thatlimT→∞⁡1N(T)∑0<Reα≤Te2iκϑ(α)=0 where ϑ(t) is the phase of ζ(12+it)=e−iϑ(t)Z(t).