On the Non-Trivial Zeros off the Critical Line for L-functions from the Extended Selberg Class
On the Non-Trivial Zeros off the Critical Line for L-functions from the Extended Selberg Class
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关于扩展 Selberg 类 L 函数的临界线外的非平凡零点
DOI：
10.1007/s00605-006-0412-x
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Mieczysław Kulas
中科院分区：
文献类型：
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作者：
J. Kaczorowski;Mieczysław Kulas
摘要。本文考虑的主要问题可以表述如下：给定一个满足黎曼假设或至少是非平凡密度估计的l函数，它是否具有欧拉积展开？一个正的答案将意味着算术是证明黎曼假设或非平凡密度估计所必需的。本文给出了扩展Selberg类的一阶l函数的一个解。证明中的主要工具是：a . Perelli和第一作者（见[7]）对扩展Selberg类的1度结构的显式描述，以及Dirichlet l -函数的“混合”型联合普惠定理。后一个结果似乎是一个独立的兴趣，体现在一个陈述的经典Kronecker-Weyl定理的丢芬图近似和沃罗宁的Dirichlet l -函数的联合普适定理的一个特例。
Abstract.The principal problem considered in this paper can be formulated as follows: given an L-function satisfying the Riemann Hypothesis or at least a non-trivial density estimate, is it true that it has an Euler product expansion? A positive answer would mean that arithmetic is necessary for proving the Riemann Hypothesis or a non-trivial density estimate, respectively. The paper contains a solution for degree one L-functions from the extended Selberg class. The main tools in the proof are: explicit description of the structure of the extended Selberg class in degree one due to A. Perelli and the first named author (see [7]) and a “hybrid” type joint universality theorem for Dirichlet L-functions. The latter result seems to be of an independent interest, embodying in one statement a special case of the classical Kronecker-Weyl theorem on diophantine approximations and Voronin’s joint universality theorem for Dirichlet L-functions.