A Discrete Limit Theorem for the Matsumoto Zeta-Function on the Complex Plane

In the paper, a discrete limit theorem in the sense of the weak convergence of probability measures for the Matsumoto zeta-function on the complex plane is proved.     

The first derivative multiple zeta values at non-positive integers

In this article, we prove some explicit results for the first derivative multiple zeta values at non-positive integers and apply them to a certain classical problem in number theory which was studied and developed by E. Hecke, A. Fujii and K. Matsumoto. Further, we consider the relation between regular values and reverse values for the multiple zeta-function via a certain functional relation.     

Universality of the Periodic Hurwitz Zeta-Function with Rational Parameter

The periodic Hurwitz zeta-function, a generalization of the classical Hurwitz zeta-function, is defined by a Dirichlet series with periodic coefficients and depends on a fixed parameter. We show that a wide class of analytic functions is approximated by shifts of a periodic zeta-function with rational parameter.     

A discrete version of the Mishou theorem

H. Mishou proved that the Riemann zeta-function and Hurwitz zeta-function with transcendental parameter are jointly universal, i.e., their shifts (continuous) approximate any pair of analytic functions. In the paper, a discrete version of the Mishou theorem is presented. In this case, the parameter of the Hurwitz zeta-function and the step of discrete shifts are connected by a certain independence relation.     

Growth of the Lerch zeta-function

It is believed that the Lindelöf hypothesis is also true for the Lerch zeta-function. Here we present results supporting this conjecture. We first consider the growth of the Lerch zeta-function assuming the generalized Lindelöf hypothesis for Dirichlet L-functions. We next prove that Huxleys exponent 32/205 in the Lindelöf hypothesis for the Riemann zeta-function holds also for the Lerch zeta-function.__________Partially supported by a grant from the Lithuanian State Science and Studies Foundation.Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 45–56, January–March, 2005.     

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.     

关于ζ函数的积分表示

由Riemannζ函数的函数方程得到Hurwitzζ函数的Hermite公式,再从Hermite公式得到Γ(s)的Binet′s第二表达式,从而由ζ函数推得Γ(s)的性质.     

On the Geil–Matsumoto bound and the length of AG codes

The Geil–Matsumoto bound conditions the number of rational places of a function field in terms of the Weierstrass semigroup of any of the places. Lewittes’ bound preceded the Geil–Matsumoto bound and it only considers the smallest generator of the numerical semigroup. It can be derived from the Geil–Matsumoto bound and so it is weaker. However, for general semigroups the Geil–Matsumoto bound does not have a closed formula and it may be hard to compute, while Lewittes’ bound is very simple. We give a closed formula for the Geil–Matsumoto bound for the case when the Weierstrass semigroup has two generators. We first find a solution to the membership problem for semigroups generated by two integers and then apply it to find the above formula. We also study the semigroups for which Lewittes’s bound and the Geil–Matsumoto bound coincide. We finally investigate on some simplifications for the computation of the Geil–Matsumoto bound.     

On the Mishou Theorem with an Algebraic Parameter

Siberian Mathematical Journal - The Riemann zeta-function and the Hurwitz zeta-function with transcendental or rational parameter are universal in the sense of Voronin: their shifts approximate...     

On the zeta-function of systems of nonlinear equations

We introduce the concept of zeta-function for a system of meromorphic functions f = (f ₁,..., f _n) in ?ⁿ. Using residue theory, we give an integral representation for the zeta-function which enables us to construct an analytic continuation of the zeta-function.     

On the Laurent coefficients of a class of Dirichlet series

The generalized Euler constants for an arithmetic progression have been considered by several authors including D. H. Lehmer, W. E. Briggs, K. Dilcher and S. Kanemitsu as an important generalization of the ordinary Euler-Stieltjes constants. In this paper, answering a problem posed by Kanemitsu, we shall adopt the partial zeta-function, a special case of the Hurwitz zeta-function, as a genuine generating function for the generalized Euler constants and make extensive use of the Hurwitz zeta-function to derive all the preceding results of Dilcher and Kanemitsu in a unified and elucidated manner.     

Nielsen zeta-function

In this paper we introduce a new zeta-function in the theory of dynamical systems. We find a sharp bound for the radius of convergence of the Nielsen zeta-function in terms of the topological entropy of the map. It follows from this that the Nielsen zeta-function has a positive radius of convergence. We prove that for an orientation-preserving homeomorphism of a compact surface the Nielsen zeta-function is either a rational function or the radical of a rational function. We calculate the Nielsen zeta-function for maps of circles, spheres, tori, protective spaces, for expanding maps of an orientable smooth compact manifold, for a homotopy periodic map of a connected compact polyhedron having no locally separating point.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 156–161, 1985.     

On the instability of the Riemann hypothesis for curves over finite fields

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) an analog of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. No number theory is required to understand the theorems and their proofs, for it is known that the zeta-functions of curves over finite fields are very explicit meromorphic functions. We study the approximation properties of these meromorphic functions.     

On a divisor problem related to the Epstein zeta-function, II

Recently by using the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and of Lü, and to establish corresponding Ω-estimates.     

Application of bounds for trigonometric sums to the problem of divisors and the mean value of the Riemann zeta-function

In this paper we give a new bound for the Riemann zeta-function in the neighborhood of the straight line =1 and indicate its application to the problem of divisors and the mean value of the Riemann zeta-function.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 539–548, November, 1972.The author is grateful to A. F. Lavrik for formulating the problem and for indicating the work of Richert.     

Mean absolute value of the Riemann zeta-function in a critical stripe

An asymptotic formula for the mean absolute value of the Riemann zeta-function in a critical stripe is obtained in the paper.     

关于广义Dedekind和的加权均值

利用Dirichlet L－函数的均值定理和特征和估计，研究了广义Dedekind和与HurwitzZeta-函数的加权均值分布性质，并给出一个有趣的渐近公式。     

On a Joint Limit Distribution of the Riemann Zeta-function in the Space of Analytic Functions

In the paper, the explicit form of the limit measure in a joint limit theorem for the Riemann zeta-function in the space of analytic functions is given.     

On Joint Universality of the Riemann Zeta-Function

Mathematical Notes - A theorem is obtained on the approximation of a collection of analytic functions in short intervals by a collection of shifts of the Riemann zeta-function...     

Generalization of the Plana Formula

We obtain an analog of the Plana formula, which is essential in obtaining the functional equation for the classical Riemann zeta-function.