Related papers: Numerical computations on the zeros of the Euler d…
We study the behavior of zero-divisors of the double zeta-function $\zeta_2(s_1,s_2)$. In our former paper \cite{MatSho14} we studied the case $s_1=s_2$, but in the present paper we consider the more general two variable situation. We carry…
The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which…
In this paper, we give Hurwitz zeta distributions with $0 < \sigma \ne 1$ by using the Gamma function. During the proof process, we show that the Hurwitz zeta function $\zeta (\sigma,a)$ does not vanish for all $0 <\sigma <1$ if and only if…
We investigate the statistical distribution of the zeros of Dirichlet $L$--functions both analytically and numerically. Using the Hardy--Littlewood conjecture about the distribution of prime numbers we show that the two--point correlation…
The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of…
In this paper, we give Hurwitz-Lerch zeta distributions with $0 < \sigma \ne 1$ by using the Gamma function. Moreover, we define Hurwitz-Lerch type of Euler-Zagier double zeta distributions not only in the region of absolute convergence but…
We investigate the distribution of the zeros of partial sums of the Riemann zeta-function, sum_{n\leq X}n^{-s}, estimating the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of…
Numerical investigations around a transformation of Landau's formula suggest certain statistical regularities in the distribution of zeros of the Riemann zeta function.
For the multiple zeta function zeta2(s1,s2) of two variables,we obtain its integral representation(involving product of Hurwitz zeta functions) over the interval [1,infinity),with respect to second variable of Hurwitz zeta function and also…
This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…
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…
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
We first give a condition on the parameters $s,w$ under which the Hurwitz zeta function $\zeta(s,w)$ has no zeros and is actually negative. As a corollary we derive that it is nonzero for $w\geq 1$ and $s\in(0,1)$ and, as a particular…
We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $\zeta(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with…
We introduce multiple versions of L-functions for Witten zeta functions. We study their algebraic and analytic properties. Especially we investigate the existence of zeros at negative integers. These results strongly suggest the universal…
We prove asymptotic formulas for mean square values of the Euler double zeta-function $\zeta_2(s_0,s)$, with respect to $\Im s$. Those formulas enable us to propose a double analogue of the Lindel{\"o}f hypothesis.
In this paper we study the higher-order Euler numbers and polynomials and we introduce the mutiple zeta functions which interpolate higher-order Euler polynomials and numbers at negative integers
As we have shown several years ago [Y2], zeros of $L(s, \Delta )$ and $L^(2)(s, \Delta )$ can be calculated quite efficiently by a certain experimental method. Here $\Delta$ denotes the cusp form of weight 12 with respect to SL$(2, Z)$ and…
For an arbitrary complex number $a\neq 0$ we consider the distribution of values of the Riemann zeta-function $\zeta$ at the $a$-points of the function $\Delta$ which appears in the functional equation $\zeta(s)=\Delta(s)\zeta(1-s)$. These…
We give results on zeros of a polynomial of $\zeta(s),\zeta'(s),\ldots,\zeta^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate…