We study distributions of differences of unscaled Riemann zeta zeros, $γ-γ^{'}$, at large. We show, that independently of the location of the zeros, i.e., even for zeros as high as $10^{23}$, their differences have similar statistical properties. The distributions of differences are skewed usually towards the nearest zeta zero. We show, however, that this is not always the case, but depends upon the distance and number of nearby zeros on each side of the corresponding distribution. The skewness, however, always decreases when zeta zero is crossed from left to right, i.e., in increasing direction. Furthermore, we show that the variance of distributions has local maximum or, at least, a turning point at every zeta zero, i.e., local minimum of the second derivative of the variance. In addition, it seems that the higher the zeros the more compactly the distributions of the differences are located in the skewness-kurtosis -plane. Furthermore, we show that distributions can be fitted with Johnson probability density function, despite the value of skewness or kurtosis of the distribution.
Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential equation, involving an infinite series of fractional derivatives, which is satisfied by the Lerch zeta function.
We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any "natural" sequence of zeta functions which conver…
We study distributions of differences of unscaled Riemann zeta zeros, $γ-γ^{'}$, at large distances. We show, that independently of the height, a subset of finite number of successive zeros knows the locations of lower level zeros. The information contained in the subset of zeros is inversely proportional to $ln(γ/(2π))$, where $γ$ is the average zeta of the subset. Because the mean difference of the zeros also decreases as inversely proportional to $ln(γ/(2π))$, each equally long segment of the line $\Re(z)=1/2$ contains equal amount of information. The distributions of differences are skewed…
We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical "repulsion" between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarit…
Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field
研究了在大亏格极限下，定义在固定有限域上的一类超椭圆曲线的zeta函数零点分布的涨落。根据曲线的黎曼假设，零点都在一个圆上。它们的角是均匀分布的，所以对于亏格为g的曲线，随着亏格的增长，固定的区间I将包含2g个|I|角。证明了当I中角数的方差渐近为log（2 g |I|）的常数倍时，正则化的涨落是高斯的，并证明了一个中心极限定理.只要角2g |I|的期望数目趋于无穷大，这些结果对于收缩区间继续保持。
Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the corresponding zeta zeros. In this paper, we provide a model that estimates how often this pattern occurs. The model is based on a conjecture that relates the statistical distribution of the zeta zeros to that of the eigenvalues of random unitary matrices.
We study distributions of differences of unscaled Riemann zeta zeros, $γ-γ'$, at large. We show, that independently of the location of the zeros, their differences have similar statistical properties. The distributions of differences are skewed towards the nearest zeta zero, have local maximum of variance and local minimum of kurtosis at or near each zeta zero. Furthermore, we show that distributions can be fitted with Johnson probability density function, despite the value of skewness or kurtosis of the distribution.
A synthetic algebraic approach to a discussion of Riemann's hypothesis
一个新的方法来长期争论的问题，提出了从GRAM-EQULUND分析连续的Zeta函数（G-B Zeta表达式）。考虑到对称（偶指数）和反对称（奇指数）的幂级数表示的G-B制定沿圆形路径包含4个假设的零点HADAMARD和DE LA VALLEE POUSSIN预测（H/DLVP“外围”四重奏）。从假设四重态的代表零点处的偶指数分量和奇指数分量所需的必要条件出发，导出了一些有趣的逻辑结论，并在黎曼假设的框架下作了简要的讨论.将假设的四重态的代表零点暂时映射到一个更高维的辅助域上，为关于存在这种外围零点的可能性的否定结论提供了一条有趣的捷径。
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…
The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n α)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Here $α$ is a real parameter, and $(a_n)_{n \geq 1}$ is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number $α$, in terms of the additive energy of the in…
On Hilbert-Polya conjecture: Hermitian operator naturally associated to L-functions
Using as starting point a classical integral representation of a L-function we define a familly of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This Hermitian operator can take also other forms, more symetric. In the case of particular L-functions, like Zeta function or Dirichlet L-functions, the eigenfunctions defined for this operator have symmetry properties. Moreover, for s zero fo Zeta function (or Dirichlet L-function), the associated eigenfunction has a specific property (a part of eigenfunction is can…
We study the Epstein zeta-function formulated on the $d$-dimensional hypercubic lattice, $ζ^{(d)}(s) = (1/2)\sideset{}{'}\sum_{n_1,\ldots,n_d} (n_1^2+\cdots+n_d^2)^{-s/2}$ where $\Re(s)>d$ and the summation runs over all integers excluding the origin. An analytical continuation of the Epstein function to the whole complex $s$-plane is constructed for spatial dimension $d$ being a continuous variable ranging from $0$ to $\infty$. We are interested in zeros $ρ=ρ_x+{\rm i}ρ_y$ defined by $ζ^{(d)}(ρ) = 0$. Besides the trivial zeros, there exist "critical" zeros (on the critical line) with $ρ_x=\fr…
Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals, which are known lie on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$, our current work recovers that of Armitage, which from th…
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical numb…
The Alternative Hypothesis 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 nontrivial 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 counte…
