Related papers: Numerical computations on the zeros of the Euler d…

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the…

We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of $\zeta$--functions at the negative integers. By using formal calculus techniques we…

We study analytic properties of the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with a transcendental parameter. We first survey briefly previous results, and then…

We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating…

We study the limit distribution of eigenvalues of a Ruelle operator (which is also called the Thurston pushforward operator) for the dynamical system $z \mapsto z^2+c$ when $c<-2$ and tends to $-2$.

The zeta functions for the Schr\"odinger equation with a triangular potential are investigated. Values of the zeta functions are computed using both the Weierstrass factorization theorem and analytic continuation via contour integration.…

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…

We introduce and study "elliptic zeta values", a two-parameter deformation of the values of Riemann's zeta function at positive integers. They are essentially Taylor coefficients of the logarithm of the elliptic gamma function, and share…

For the higher order derivative(with respect to the first variable) of Hurwitz zeta function,we discuss as a function of the second variable,the location and the nature of its singularities and obtain the formulae for its derivative and…

A route to the derivation of the numbers $s$ to the transcendental equation $\zeta(s)=0$ is presented. The solutions to this equation require the solving of a geodesic flow in an infinite dimensional manifold. These solutions enable one…

The Riemann hypothesis, which states that the non-trivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. Inspired by the P\'olya-Hilbert…

We give an informal survey of the historical development of computations related to prime number distribution and zeros of the Riemann zeta function.

Let $a, w_1, w_2,\cdot\cdot\cdot, w_r >0$ and $s \in \mathbb{C}$. We put $w= (w_1,\cdot\cdot\cdot,w_r)$. Then the Barnes $r$-ple zeta function is defined by $\zeta_r(s, w, a) = \sum_{m_1=0}^{\infty} \cdot\cdot\cdot \sum_{m_r=0}^{\infty}…

Numerical investigations around a transformation of Landau's formula suggest certain statistical regularities in the distribution of zeros of the Riemann zeta function.

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product…

The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form…

The aim of the present paper is to study the relations between the prime distribution and the zero distribution for generalized zeta functions which are expressed by Euler products and is analytically continued as meromorphic functions of…

The Riemann Zeta function $\zeta(s)$ never vanishes in the region : $$ \Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2). $$

For an analytic family P_s of polynomials in n variables (depending on a complex number s, and defined in a neighborhood of s = 0), there is defined a monodromy transformation h of the zero level set V_s= {P_s=0} for s different from 0,…

Let $X$ be a smooth proper curve over a finite field and let $\infty \in X$ be a closed point. Let $A$ be the ring of functions on $X - \infty$. The Goss zeta function $\zeta_A$ of $A$ is an equicharacteristic analogue of the Riemann zeta…