Analytic properties of Dirichlet generating functions of arithmetic objects
Analytic properties of Dirichlet generating functions of arithmetic objects
复制标题
算术对象狄利克雷生成函数的解析性质
DOI：
10.1007/bf01140029
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发表时间：
1978
期刊：
影响因子：
--
通讯作者：
S. Voronin
中科院分区：
文献类型：
--
作者：
S. Voronin
本文研究了数论中一些问题中出现的黎曼函数以及与其类似的函数。黎曼在 [I] 中强调了研究狄利克雷生成函数的解析性质（特别是研究黎曼 E 函数的零值分布）的重要性。黎曼函数的零值分布首先由 H. Bohr 提出。例如，对他而言，以下定理（参见[2, 5]）：固定〜E（I / 2，i]和t从--〜到+〜变化的值的集合〜（〜+ it）在C中处处稠密。玻尔给出的该定理的证明使用素数乘积形式的〜（s）的欧拉表示，素数对数的独立性以及克罗内克近似后来，类似于玻尔的论证被应用于获得 I~(~~ it) l 增长率的下界，即黎曼 E 函数理论的所谓 m 定理。
In the dissertation we examine the Riemann~-function and functions similar to it, arising in some problems in number theory. The importance of studying the analytic properties of Dirichlet generating functions (in particular, of studying the distribution of zero values of the Riemann E-function) was stressed by Riemann in [I]. The distribution of the zero values of the Riemann~-function was first taken up by H. Bohr. To him is due, for instance, the following theorem (see [2, 5]): the set of values of~(~+ it) for fixed~ E (I/2, i] and for t varying from--~ to+~ is everywhere dense in C. The proof of this theorem, given by Bohr, used the Euler representation of~(s) in the form of a product by primes, the independence of the logarithms of primes, and the Kronecker approximation theorem. Later on, arguments analogous to those of Bohr were applied to obtain lower bounds of the rate of growth of I~(~~ it) l, ie, in the so-called m-theorems of the theory of the Riemann E-function.