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.     

On mixed joint discrete universality for a class of zeta-functions. II

We investigate the mixed joint discrete value distribution and the mixed joint discrete universality for the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with transcendental parameter. The common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are used.

     

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...     

Limit theorems in the space of analytic functions for the Hurwitz zeta-function with an algebraic irrational parameter

We prove a limit theorem in the space of analytic functions for the Hurwitz zeta-function with algebraic irrational parameter.     

A limit theorem for the Hurwitz zeta-function with an algebraic irrational parameter

We prove a limit theorem in the complex plane for the Hurwitz zeta-function with algebraic irrational parameter. Received: 14 June 2004; revised: 12 April 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.     

Discrete value distribution of the Hurwitz zeta-function with algebraic irrational parameter. I

In the paper, the statistical properties of discrete values of the Hurwitz zeta-function with algebraic irrational parameter in the space of analytic functions are investigated.     

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.     

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 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.     

On the Mean Value of the Dedekind Sum with the Weight of Hurwitz Zeta-Function

The main purpose of this paper is to use the mean value theorem of the Dirichlet L-functions to study the asymptotic behavior of the Dedekind sums with a weight of Hurwitz zeta-function and to give an interesting asymptotic formula.     

A hybrid mean value formula involving Kloosterman sums and Hurwitz zeta-function

The main purpose of this paper is using the mean value theorem of Dirichlet L-function and the estimates for character sums to study the asymptotic properties of a hybrid mean value of Kloosterman sums with the weight of Hurwitz zeta-function and the Cochrane sums, and give an interesting mean value formula for it.     

On limit distribution of the Hurwitz zeta-function

The distribution of the vector (|ζ(s, α)|; ζ(s, α)), where ζ(s, α) is the Hurwitz zeta-function with transcendental parameter α, is considered and a probabilistic limit theorem is obtained. Also, the dependence between |ζ(s, α)| and ζ(s, α) in terms of m-characteristic transforms is discussed.     

The Atkinson formula for the periodic zeta-function

We give an Atkinson-type formula for the periodic zeta-function ζ_λ with rational parameter λ. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 504–516, October–December, 2007.     

Some value-distribution theorems for periodic Hurwitz zeta-functions

This present paper is a survey of recent value-distribution results for periodic Hurwitz zeta-functionsis. These results include probabilistic limit theorems, universality, and the functional independence of periodic Hurwitz zeta-functions. Both one-dimensional and joint value-distribution theorems are concerned.     

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 real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta-function

     

A note on the Selberg class (degrees 0 and 1)

We give a classification of the Dirichlet polynomials in the Selberg class as entire quotients of Dirichlet series with periodic coefficients and the Riemann zeta-function. Received: 2 November 2001     

Universality theorems for zeta-functions with periodic coefficients

We obtain Voronin-type universality theorems for some classes of functions of a collection of periodic zeta-functions and periodic Hurwitz zeta-functions with algebraically independent parameters.     

On Statistical Properties of the Lerch Zeta-Function. II

A discrete limit theorem for the Lerch zeta-function with an integer parameter in the space of meromorphic functions is proved.