Value-distribution of zeta-functions
Value-distribution of zeta-functions
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zeta 函数的值分布
DOI：
10.1007/bfb0097134
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
Kohji Matsumoto
中科院分区：
文献类型：
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作者：
Kohji Matsumoto
本文的目的是给出与zeta函数的值分布有关的渐近概率测度的存在性的迄今为止最简单的证明。渐近测度的存在性定理首先是由Bohr-Schwarzen [2]在Riemann zeta-函数的情况下得到的，其证明是基于一个关于凸曲线和的复杂理论。Eschen-Wintner [6]和Borchsenius-Eschenen [3]使用概率论中的P. Levy收敛定理给出了另一种证明。在证明中用到了凸曲线的一些性质，所以他们的方法实际上只适用于”凸”欧拉积的情形。(The我们将在最后一节解释这个术语的严格含义。然而，我们可以修改他们的概率方法来构造一个可以应用于非凸欧拉乘积的证明。这样的证明首先在[9]中针对由Heeke算子定义的zeta-函数得到。本文中的证明是[9]中的证明的简化版本，但我们将在更一般的情况下工作。
The purpose of this article is to give the hitherto simplest proof of the existence of the asymptotic probability measure connected with the value-distribution of zeta-functions. The existence theorem of the asymptotic measure was first obtained by Bohr-Jessen [2] in the case of the Riemann zeta-function, whose proof is based on a complicated theory on the sums of convex curves. Jessen-Wintner [6] and Borchsenius-Jessen [3] have shown an alternative proof, using P. Levy's convergence theorem in probability theory. Some properties of convex curves are used in their proof, so their method, as it is, can be applied only to the case of" convex" Euler products.(The rigorous meaning of this term we will explain in the last section.) However, we can modify their probabilistic approach to construct a proof which can be applied to non-convex Euler products. Such a proof was first obtained in [9] for zeta-functions defined by Heeke operators. The proof in the present paper is a simplified version of the one in [9], but we will work in a more general situation.