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     

Limit theorems for the Estermann zeta-function. II

A limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function is obtained. Partially supported by Lithuanian Foundation of Studies and Science     

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.     

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.     

Some limit theorems for a Dirichlet series related to the Euler function

Using some conjectures on the mean-value of generalized numbers, limit theorems in the sense of the weak convergence of probability measures for the Dirichlet series related to the Euler function are proved. Also, the limit measure is studied. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania: Šiauliai University, P. Višinskio 25, 5400 Šiauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 331–342, July–September, 1999. Translated by A. Laurinčikas     

Joint Discrete Limit Theorems on the Complex Plane for General Dirichlet Series

We prove a joint discrete limit theorem in the sense of weak convergence on the complex plane for general Dirichlet series.     

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。     

A Limit Theorem for Dirichlet L-Functions

We prove a limit theorem on the weak convergence of probability measures in the space of continuous functions for Dirichlet L-functions. The result generalizes a similar theorem for the Riemann zeta-function.     

A Joint Limit Theorem on the Complex Plane for General Dirichlet Series

We prove a joint two-dimensional limit theorem in the sense of weak convergence of probability measures on the complex plane for general Dirichlet series.     

Value Distribution of General Dirichlet Series. V

We prove a joint universality theorem for general Dirichlet series. The conditions of this theorem are weaker than those in [7].     

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.     

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.     

Functional limit theorems for Levy processes and their almost-sure versions

In this paper, we prove a criterion of convergence in distribution in the Skorokhod space. We apply this criterion to some special Levy processes and obtain almost-sure versions of limit theorems for these processes. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 81–92, January–March, 2007.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.     

Value Distribution of General Dirichlet Series. IV

In the paper, a limit theorem in the space of meromorphic functions with explicitly given limit measure for general Dirichlet series is proved. The theorem obtained generalizes and simplifies the results of work [7].     

On fractional moments of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions. III

In this paper, we consider upper and lower bounds of the same order with explicitly given constants for fractional moments of Dirichlet L-functions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 280–295, April–June, 2007.     

Weak convergence of Dirichlet processes

Weak convergence of Markov processes is studied by means of Dirichlet forms and two theorems for weak convergence of Hunt processes on general metric spaces are established. As applications, examples for weak conver gence of symmetric or non-symmetric Dirichlet processes on finite and infinite spaces are given. Project partially supported by the National Natural Science Foundation of China and Tianyuan Mathematics Foundation.     

On special values of certain Dirichlet L-functions

Let r _k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r ₉(t), t≡5 (mod 8), r ₁₁(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r ₉(t), r ₁₁(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.      

Dirichlet Series Associated with Strongly q-Multiplicative Functions

In this work the Dirichlet series associated with real strongly q-multiplicative functions f(n) are studied. We will confine ourselves to the case _i=0 ^q–1 f(i) = 0. It is known that in this case the function _f (s) has an analytic continuation to the whole complex plane as an entire function with trivial zeros on the negative real line. The real function _f (t) satisfying the integral equation with delayed argument for some nonzero real _f naturally appears in the representation of the function _f (s). In this article we find some asymptotic properties of the function _f (s), prove that _f (s) is an entire function of order 2, and also prove that in the region the function _f (s) has only trivial zeros which are simple.