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.     

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.     

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 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.     

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.     

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     

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 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 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.     

关于ζ函数的积分表示

主要研究了ζ函数的积分表示形式;通过解析数论的研究方法,利用黎曼ζ函数方程,给出了关于赫尔维茨ζ函数的埃尔米特公式,利用埃尔米特公式得出关于Γ函数的比内第二表达式,通过ζ函数得出Γ函数一些性质.     

Равномерные приближ ения и нули в коротких интервалах производныхZ-функц ии Харди

Approximate functional equations, uniform in their order, are obtained for the derivatives of Riemann's zeta-function on the critical line and for those of Hardy's function. Relying on them, a new theorem is proved on the zeros of the derivatives of Hardy's function on short intervals.     

A note on sums of products of Bernoulli numbers

In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function.     

On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta-function

     

Inequalities involving the multiple psi function

In this work, multiple gamma functions of order n have been considered. The logarithmic derivative of the multiple gamma function is known as the multiple psi function. Subadditive, superadditive, and convexity properties of higher-order derivatives of the multiple psi function are derived. Some related inequalities for these functions and their ratios are also obtained.     

On an algebraic version of the Knizhnik–Zamolodchikov equation

A difference equation analogue of the Knizhnik?CZamolodchikov equation is exhibited by developing a theory of the generating function H(z) of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which H(z) is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples of non-positive integers. A generating function for these polynomials satisfies a similar difference equation to that of H(z), but in contrast to the fact that said polynomials have rational coefficients, the algebraic independence of the usual Hurwitz zeta functions is proven, and the Hurwitz polyzeta functions are shown to satisfy no algebraic relations other than those arising from the shuffle relations. The values of the NMBPs at z=1 provide a regularization of the multiple zeta values at tuples of negative integers, which is shown to agree with the regularization given in Akiyama et al. (Acta Arith. 98:107?C116, 2001). Various elementary properties of these values are proven.     

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.