On Bohr's theorem for multiple Fourier series

The problem of Pringsheim uniform convergence of multiple Fourier series in the trigonometric system is considered. A multidimensional analog of Bohr's theorem on the uniform convergence of the Fourier series of a continuous function after a homeomorphic chance of variable is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 913–924, December, 1998.     

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [5].     

A mean value theorem on Dirichlet series

This paper deals with mean-value for the square of certain functionF(s) which has some characteristic properties of the Riemann zeta-function and its powers.Work supported by the University of the Basque Country.     

The differentiation operator in the space of uniformly convergent Dirichlet series

Continuity, compactness, the spectrum and ergodic properties of the differentiation operator are investigated, when it acts in the Fréchet space of all Dirichlet series that are uniformly convergent in all half-planes for each . The properties of the formal inverse of the differentiation are also investigated.     

On the sign changes of coefficients of general Dirichlet series

Under what conditions do the (possibly complex) coefficients of a general Dirichlet series exhibit oscillatory behavior? In this work we invoke Laguerre's Rule of Signs and Landau's Theorem to provide a rather simple answer to this question. Furthermore, we explain how our result easily applies to a multitude of functions.

     

Joint discrete limit theorems in the space of analytic functions for general Dirichlet series

We prove joint discrete limit theorems in the sense of weak convergence of probability measures in the space of analytic functions for general Dirichlet series.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 104–116, January–March, 2005.Translated by R. Macaitien     

Value distribution of general Dirichlet series. VII

We prove a limit theorem in the space of meromorphic functions for a new class of general Dirichlet series. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 193–202, April–June, 2006.     

Exact estimates for integrals involving Dirichlet series with nonnegative coefficients

We consider the Dirichlet series

with coefficients for all . Among others, we prove exact estimates of certain weighted -norms of on the unit interval for any , in terms of the coefficients . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for integrals involving the former one by relying on exact estimates for integrals involving the latter one. As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series

with complex coefficients .

     

Spaces of vector-valued Dirichlet series in a half plane

Dirichlet series with real frequencies which represent entire functions on the complex plane C have been investigated by many authors. Several properties such as topological structures, linear continuous functionals, and bases have been considered. Le Hai Khoi derived some results with Dirichlet series having negative real frequencies which represent holomorphic functions in a half plane. In the present paper, we have obtained some properties of holomorphic Dirichlet series having positive exponents, whose coefficients belong to a Banach algebra.     

Dirichlet series of Rankin-Cohen brackets

Given modular forms f and g of weights k and ?, respectively, their Rankin-Cohen bracket corresponding to a nonnegative integer n is a modular form of weight k+?+2n, and it is given as a linear combination of the products of the form f(r)g(n−r) for 0?r?n. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.     

Formulas for the Riemann zeta-function and certain Dirichlet series

Using known theta identities and formulas of S. Ramanujan and G. Hardy among others we prove several formulas for the Riemann zeta-function and two Dirichlet series.     

A Joint Limit Theorem for General Dirichlet Series

In this paper, a joint limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for general Dirichlet series is proved. The explicit form of the limit measure is given.     

On the asymptotic independence of Dirichlet series

The joint limit distribution of functions given by Dirichlet series is studied. The necessary and sufficient condition when this distribution is a product of marginal distributions is found. An example of such Dirichlet series with linear independent systems of exponents is presented. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius; Šiauliai University, P. Višinskio 25, 5419 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 65–73, January–March, 1999. Translated by A. Laurinčikas     

On the Maximum of the Modulus and the Maximal Term of Dirichlet Series

For the Dirichlet series, we obtain a condition on the exponents for which the logarithms of the maximum of the modulus of its sum and of the maximal term are equivalent to the same convex function.     

A Joint Limit Theorem for General Dirichlet Series

We obtain a joint limit theorem in the sense of weak convergence of probability measures for general Dirichlet series in the space of meromorphic functions.     

全平面内随机Dirichlet级数的几个定理

利用Nevanlinna 理论研究了全平面内随机Dirichlet 级数所表示的整函数的增长性和值分布,得到全平面内水平带形上的几个新的定理,推广了以往研究半平面内水平半带形上关于增长性和值分布的一些相关结论。     

Discrete Limit Theorems for General Dirichlet Series. II

We prove a discrete limit theorem for general Dirichlet series in the sense of weak convergence of probability measures in the space of analytic funtions. Examples are presented.