The universality of the Lerch zeta-function
The universality of the Lerch zeta-function
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Lerch zeta 函数的普遍性
DOI：
10.1007/bf02465359
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发表时间：
1997
影响因子：
0.4
通讯作者：
A. Laurinčikas
中科院分区：
文献类型：
--
作者：
A. Laurinčikas
设s= cr+ it是一个复变量。Lerch zeta函数L（~.，t，s），r> 1，由Dirichlet级数e2 JriL m L（L，or，s）=（m+ ct）s' rn= 0定义，否则由解析延拓定义。来吧~。和a是真实的数，0< ot~< 1。如果3. r Z，其中Z表示所有整数的集合，则LQ，或者，s）是整个函数。在这种情况下，不失一般性，我们可以假设0<~，< 1。文[9-11]证明了Lerch zeta函数概率测度弱收敛的几个极限定理。本文利用解析函数空间中的一个极限定理证明了L（L，c~，s）的普适性。回想一下，黎曼zeta函数的普适性是由SM Voronin [14]发现的。设K是一个半径r< 1/4的闭圆盘，中心为s= 3/4，函数f（s）在K上连续非零，在K内部解析. Voronin定理断言，对于每一个e> 0，存在一个真实的数r，使得maxl ∈（s+ it）-f（s）[< E]。瑞典克朗
Let s= cr+ it be a complex variable. The Lerch zeta-function L (~., ot, s) for, r> 1 is defined by the Dirichlet series e2JriL m L (L, or, s)=(m+ ct) s' rn= O and otherwise by analytic continuation. Here~. and a are real numbers, 0< ot~< 1. If 3. r Z, where Z denotes the set of all integer numbers, then LQ., or, s) is an entire function. In this case, without loss of generality we can assume that 0<~,< 1. In [9-11], we proved several limit theorems concerning the weak convergence of probability measures for the Lerch zeta-function. The aim of this note is to use a limit theorem in the space of analytic functions for the proof of the universality of L (L, c~, s). Recall that the universality of the Riemann zeta-function~'(s) was discovered by SM Voronin [14]. Let K be a closed disk of radius r< 1/4 centered at s= 3/4, and let the function f (s) be continuous and nonvanishing on K and analytic in the interior of K. The Voronin theorem asserts that for every e> 0 there exists a real number r such that maxl~'(s+ it)-f (s)[< E. sEK