We prove three results on the density resp. local density and clustering of zeros of the Beurling zeta function $ζ(s)$ close to the one-line $σ:=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. Theorem 4 provides a zero density estimate, which is a complement to known results for the Selberg class. Note that density results for the Selberg class rely on use of the functional equation of $ζ$, which we do not assume in the Beurling context. In Theorem 5 we deduce a variant of a well-known theorem of Turán, extending its range of validity even for rectangles of height only $h=2$. In Theorem 6 we will extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result -- which, on the other hand, is a strong sharpening of the average result from the classic book \cite{Mont} of Montgomery -- was worked out by Diamond, Montgomery and Vorhauer. Here we show that the obscure technicalities of the Ramachandra paper (like a polynomial with coefficients like $10^8$) can be gotten rid of, providing a more transparent proof of the validity of this clustering phenomenon.
We show that the analytic continuations of Helson zeta functions $ ζ_χ(s)= \sum_1^{\infty}χ(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis.
The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s).
For a function field $k$ over a finite field with $\mathbb{F}_q$ as the field of constant, and a finite abelian group $G$ whose exponent is divisible by $q-1$, we study the distribution of zeta zeroes for a random $G$-extension of $k$, ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution.
This article considers the positive integers $N$ for which $ζ_{N}(s) = \sum_{n=1}^{N} n^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on earlier results, we show that there are no zeroes for $1\leq N\leq 18$ and for $N=20, 21, 28$. For all other $N$ there are infinitely many zeroes.
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-π,π)$ has a standard Gaussian distribution (when properly normalized).
Continuing previous study of the Beurling zeta function, here we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $ρ_0$ of the Beurling zeta function $ζ_P$ for a given number system generated by the primes $P$, the corresponding error term $Δ_P(x):=ψ_{P}(x)-x$, where $ψ_{P}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough…
We use visible point vector identities to examine polylogarithms in the neighbourhood of the Riemann zeta function zeroes. New formulas limiting to the trivial zeroes and to the critical line on the zeta function are given. Similar results arise from Euler-Zagier sums given in Bailey, Borwein and Crandall [4] providing new infinite product identities.
This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture. We show that $\limsup M(x)x^{-1/2} \geq 1.6383$ and that $\liminf M(x)x^{-1/2}\leq -1.6383$.
We study generalised prime systems $\mathcal{P}$ $(1<p_1\leq p_2\leq...,$ with $p_j\in\R$ tending to infinity) and the associated Beurling zeta function $ζ_{\mathcal{P}}(s) =\prod_{j=1}^{\infty} (1-p_j^{-s})^{-1}$. Under appropriate assumptions, we establish various analytic properties of $ζ_{\mathcal{P}}(s)$, including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of $ζ_{\math…
We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $ζ(s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $ζ(s)$ on the critical line and $h \ll 1 / \log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $ζ(s)$ of a given multiplicity.
On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.
The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0,\infty)$, which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors $θ_k\in(0,1)$, $k\ge1$. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $θ_k$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion. We start here the study of these criteria, with a special focus on exponential and gamma distributions. The m…
We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line ${Re}(s)=\frac12$ and Blaschke zeta zeroes.
We present the first applications of the recently established by us (arXiv:1304.7895; Ukrainian Math. J. - 2014. -66. - P. 371-383) generalized Li's criterion equivalent to the Riemann Hypothesis. This criterion is the statement that the Riemann hypothesis is equivalent to the non-negativity of the derivatives 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln(ξ(z))) for z=b+1 of the Riemann xi-function for all real b>-1/2 and all m = 1, 2, 3... We show that for any positive integer n there is such value of b_n (depending on n) that for all m<=n and b>b_n, inequality 1/((m-1)!)*d^m/dz^m((z+b)^(m-1)*ln(ξ(z))) …
On the assumption of the Riemann hypothesis, we show that over a class of sufficiently smooth test functions, a measure conjectured by Bogomolny and Keating coincides to a very small error with the actual pair correlation measure for zeroes of the Riemann zeta function. Our result extends the well known result of Montgomery that over the same class of test functions the pair correlation measure coincides (to a larger error term) with that of the Gaussian Unitary Ensemble (GUE). The restriction of test functions remains stringent, but we are nonetheless able to detect, at a microscopically blur…
In this paper we work out a Riemann-von Mangoldt type formula for the summatory function $ψ(x):=\sum_{g\in G, |g|\le x} Λ_{G}(g)$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $Λ_{G}$ is the corresponding von Mangoldt function attaining $\log|p|$ for $g=p^k$ with a prime element $p\in G$ and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function $ζ_{G}$, belonging to $G$, to the number of zeroes of $ζ_G$ in various regions, in particular within the critical strip where the analytic continuation exists…
We look at approximations $ ζ_X $ of the $ζ$-function introduced in Gonek's paper (arXiv:0704.3448). We look at how close the approximate zeroes are to the actual zeroes when (i) X is fixed (Section 1)(ii) X varies like $ t/2π$ (Section 3.1). We establish a heuristic for estimating these differences, involving values of $ F_X^\star(t) $ and its near-constant slopes near zeta-zero ordinates $ γ$. In Section 3.2 we see the slope around the zeroes behaves logarithmically and we calculate a numerical formula for it. In Section 3.3 and 3.4 we scale the differences with the slopes and compare them w…
Thanks to earlier work of Koiran, it is known that the truth of the Generalized Riemann Hypothesis (GRH) implies that the dimension of algebraic sets over the complex numbers can be determined within the polynomial-hierarchy. The truth of GRH thus provides a direct connection between a concrete algebraic geometry problem and the P vs.NP Problem, in a radically different direction from the geometric complexity theory approach to VP vs. VNP. We explore more plausible hypotheses yielding the same speed-up. One minimalist hypothesis we derive involves improving the error term (as a function of the…
Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory classical functions which are closely linked to zeta zeroes like psi(x), theta(x), pi(x) or the k-th prime number.
We establish a uniform upper estimate for the values of zeta(s)/zeta(s+A), 0<= A, on the critical line (conditionally on the Riemann Hypothesis). We use this to give a variant, purely complex analytic, to Baez-Duarte's proof of a strengthened Nyman-Beurling criterion for the validity of the Riemann Hypothesis.
We prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. These are concerned with the question of Ingham who asked for optimal and explicit order estimates for the error term $Δ(x):=ψ(x)-x$, given any zero-free region $D(η):=\{s=σ+it\in\mathbb{C}~:~ σ:=\Re s \ge 1-η(t)\}$. In the classical case essentially sharp results are due to some 40 years old work of Pintz. Here we consider a given a system of Beurling primes $\mathcal{P}$, the generated arithmetical semigroup $\mathcal{G}$ and the corresponding …
We slightly improve the lower bound of Baez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called `Hilbert-Polya idea'.
In math.NT/9907019 we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on the work of Wan, Diaz-Vargas, Thakur, Poonen, and Sheats for the zeta function $ζ_{\Fr[θ]}(s)$. During the writing of math.NT/9907019, we made two assumptions that have subsequently proved to be incorrect. The first assumption is that we can ignore the trivial zeroes of characteristic p L-series in formulating our conjectures. Instead, we show here how the trivial zeroes influence nearby zeroes and so lead to counter-examples of the original Riemann hypothesis analog. We t…
We define, answering a question of Sarnak in his letter to Bombieri, a symplectic pairing on the spectral interpretation (due to Connes and Meyer) of the zeroes of Riemann's zeta function. This pairing gives a purely spectral formulation of the proof of the functional equation due to Tate, Weil and Iwasawa, which, in the case of a curve over a finite field, corresponds to the usual geometric proof by the use of the Frobenius-equivariant Poincaré duality pairing in etale cohomology. We give another example of a similar construction in the case of the spectral interpretation of the zeroes of a c…
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 $\F_q$ 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 curves with $p$-rank equal to $p-1.$ While the zeta zeroes always approach the standard Gaussian distribution, the number of points over $\F_q$ has a distribution that varies with the specific family.
The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of Riemann's zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large scale high accuracy numerical experiments. So far no theoretical explanation to the observed phenomena was found.
We show that the analytic continuations of Helson zeta functions $ ζ_χ(s)= \sum_1^{\infty}χ(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole critical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis for the function $ χ$ taking values in cubic roots of unity. If the sets are symmetric with respect to the real axis, the same can be achieved with $ χ$ taking values $ \pm 1 $.
We have studied some properties of the special Gram points of the Riemann zeta function which lie on contour lines ${\bf Im}(ζ( s )) = 0$ which do not contain zeroes of $ζ( s )$. We find that certain functions of these points, which all lie on the critical line ${\bf Re}( s ) = 1/2$, are correlated in remarkable and unexpected ways. We have data up to a height of $t = 10^4$, where $s = σ+ it$.
Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure μ on $\mathbb{R}$, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments o…
In this paper we study asymptotic properties of families of zeta and $L$-functions over finite fields. We do it in the context of three main problems: the basic inequality, the Brauer--Siegel type results and the results on distribution of zeroes. We generalize to this abstract setting the results of Tsfasman, Vlăduţ and Lachaud, who studied similar problems for curves and (in some cases) for varieties over finite fields. In the classical case of zeta functions of curves we extend a result of Ihara on the limit behaviour of the Euler--Kronecker constant. Our results also apply to $L$-functions…
In a recent article a breakthrough was made in quantum computer science which established the existence of a phenomenon commonly known as $\mathsf{MIP}^\ast = \mathsf{RE}$. We show that the existence of this phenomenon implies that the supremum of the real parts of the zeroes of the Riemann zeta function is less than one. Our main tool is an assertion about the representation theory of the rank two free group that we refer to as the charged mean ergodic theorem. Our method also involves the construction of novel modular functions we refer to as complete electromagnetic functions.
We generalize the first part of A. Connes paper (math/9811068) on the zeroes of the Riemann zeta function from a number field $k$ to any simple algebra $M$ over $k$. To a given automorphic representation $π$ of the reductive group $M^\times$ of invertible elements of $M$ we find a Hilbert space $H_π$ and an operator $D_π$ (Polya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of $D_π$ coincides with the purely imaginary zeroes of the function $L(π,\rez{2} +z)$. As a byproduct we get holomorphicity of all automorphic $L$-functions, not only the…
We prove that there is a prime between $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.217))$. Our new tool which we derive is a version of Landau's explicit formula for the Riemann zeta-function with explicit bounds on the error term. We use this along with other recent explicit estimates regarding the zeroes of the Riemann zeta-function to obtain the result. Furthermore, we show that there is a prime between any two consecutive $m$th powers for $m \geq 4.971 \times 10^9$.
We point out analogies between (a) the explicit formulas in analytic number theory and transversal index theory, (b) Lichtenbaum's recent conjectures on special values of Hasse-Weil zeta functions and a formula for special values of Ruelle zeta functions, (c) work of Cramer on the zeroes of the Riemann zeta function and a result of Chazarain on the trace of a wave operator. These analogies suggest certain problems in the analysis on foliated spaces. The paper is mostly a review of previous work but the ideas concerning (c) are new.
This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type, which this time involve exponential functions, are established, and for the first time we have found equalities involving the integrals of the logarithm of the Riemann zeta-function taken exclusively along the real axis. Some of the equalities we hav…
For each $t \in {\bf R}$, define the entire function $$ H_t(x) := \int_0^\infty e^{tu^2} Φ(u) \cos(xu)\ du$$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_{n=1}^\infty (2π^2 n^4 e^{9u} - 3πn^2 e^{5u} ) \exp(-πn^2 e^{4u} ).$$ Newman showed that there exists a finite constant $Λ$ (the \emph{de Bruijn-Newman constant}) such that the zeroes of $H_t$ are all real precisely when $t \geq Λ$. The Riemann hypothesis is the equivalent to the assertion $Λ\leq 0$, and Newman conjectured the complementary bound $Λ\geq 0$. In this paper we establish Newman's conjecture. The argument…
Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not equal to 1/2 are non-negative if and only if the Riemann hypothesis holds true, and proved the relation k_n,a=n*(1-2a)/(n-1)!*d^n/dz^n((z-a)^(n-1)*ln(ξ(z))) taken at z=1-a. Assuming that the function /zeta(s) is non-vanishing for Re(s)>1/2+/Delta, where real 0</Delta<1/2, using this relation together with the functional equation for the /xi-function and the …
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We g…
Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now known that the truth of the Generalized Riemann Hypothesis (GRH) yields the implication that HN not in NP implies P is not equal to NP. We show that the assumption of GRH in the latter implication can be replaced by either of two more plausible hypotheses from analytic number theory. The first is an effective short interval Prime Ideal Theorem with explicit dependence …
We present computational results on the divisor class number and the regulator of a cubic function field over a large base field. The underlying method is based on approximations of the Euler product representation of the zeta function of such a field. We give details on the implementation for purely cubic function fields of signatures $(3,1)$ and $(1, 1; 1, 2)$, operating in the ideal class group and infrastructure of the function field, respectively. Our implementation provides numerical evidence of the computational effectiveness of this algorithm. With the exception of special cases, such …
The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $ζ_β(z)$ of the $β$-shift, for $β> 1$ close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in $\mathbb{F}_p$ and their factorizations, via Kronecker's Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density th…
It is well known that Euler experimentally discovered the functional equation of the Riemann zeta function. Indeed he detected the fundamental $s\mapsto 1-s$ invariance of $ζ(s)$ by looking only at special values. In particular, via this functional equation, the permutation group on two letters, $S_2\simeq\Z/(2)$, is realized as a group of symmetries of $ζ(s)$. In this paper, we use the theory of special-values of our characteristic $p$ zeta functions to experimentally detect a natural symmetry group $S_{(q)}$ for these functions of cardinality ${\mathfrak c}=2^{\aleph_0}$ (where $\mathfrak c$…
Let $<¶> \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $¶$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < ¶>: n \leq x} \frac{μ(n)}{n}$ are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to $\prod_{p \in ¶} (1 - \frac{1}{p})$ (the case when $¶$ is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of non-trivial zeroes and poles of the associated zeta…
