Joint value-distribution theorems on Lerch zeta-functions. II

We give corrected statements of some theorems from [5] and [6] on joint value-distribution of Lerch zeta-functions (limit theorems, universality, functional independence). We also present a new direct proof of a joint limit theorem in the space of analytic functions and an extension of a joint universality theorem. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 332–350, July–September, 2006.     

关于Smarandache函数与Riemann zeta-函数

对任意正整数n,著名的Smarandache函数S(n)定义为最小的正整数m使得n|m!.即S(n)=min{m∶m ∈N,n|m!).本文的主要目的是利用初等方法研究一类包含S(n)的Dirichlet级数与Riemann zeta-函数之间的关系,并得到了一个有趣的恒等式.     

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.     

Zero Point Energy of a Massless Scalar Field in the Cosmic String Space-Time

We consider the ground state energy of a massive scalar field in the space-time of a thick cosmic string in the 2+1 dimensional case for arbitrary angle deficit by using the zeta-function approach. Final numerical calculations were made in the massless case, only. We show that the zero point energy is negative, and for small angle deficit it is proportional to the fourth degree of the deficit.     

On the value distribution of the Matsumoto zeta-function

The Matsumoto zeta-function is defined by a polynomial Euler product and is a generalization of classical zeta-functions. In the paper a discrete limit theorem in the sense of the weak convergence of probability measures on the complex plane for the Matsumoto zeta-function is proved.     

On some functional relations between Mordell-Tornheim double L-functions and Dirichlet L-functions

In the past decade, many relation formulas for the multiple zeta values, further for the multiple L-values at positive integers have been discovered. Recently Matsumoto suggested that it is important to reveal whether those relations are valid only at integer points, or valid also at other values. Indeed the famous Euler formula for ζ(2k) can be regarded as a part of the functional equation of ζ(s). In this paper, we give certain analytic functional relations between the Mordell-Tornheim double L-functions and the Dirichlet L-functions of conductor 3 and 4. These can be regarded as continuous generalizations of the known discrete relations between the Mordell-Tornheim L-values and the Dirichlet L-values of conductor 3 and 4 at positive integers.     

Atkinson's formula for Hardy's function

An analogue of Atkinson's formula is proved for the integral function of Hardy's function Z(t). As an application of this formula, we analyze the behavior of the function F(T) showing that it can be approximated by a simple step-function. It follows that F(T)=O(T1/4) and F(T)=Ω_±(T1/4); these results were recently obtained by M.A. Korolev using an alternative method.     

A two-dimensional limit theorem for Mellin transforms of the Riemann zeta-function

In this paper, we obtain a two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for Mellin transforms of the square and the fourth power of the Riemann zeta-function. Partially supported by the Lithuanian Foundation of Studies and Science.     

On the Distribution of Integral Ideals and Hecke Gr(o)ssencharacters

The author uses analytic methods to study the distribution of integral ideals and Hecke Gr(o)ssencharacters in algebraic number fields.Nowak's results on the distribution of integral ideals,and Chandrasekharan and Good's results on the distribution of Hecke Gr(o)ssencharacters are improved.     

The Mean-Square of the Lerch Zeta-Function Near the Critical Line

In the paper, the asymptotics of the mean-square of the Lerch zeta-function near the critical line is obtained. Three different definitions of the closeness to the critical line are considered.