Discrepancy Estimates for the Value‐Distribution of the Riemann Zeta‐Function, IV
Discrepancy Estimates for the Value‐Distribution of the Riemann Zeta‐Function, IV
复制标题
黎曼 Zeta 函数的值分布的差异估计，IV
DOI：
10.1112/jlms/50.1.17
复制
发表时间：
1994
影响因子：
1.2
通讯作者：
Kohji Matsumoto
中科院分区：
文献类型：
--
作者：
G. Harman;Kohji Matsumoto
在一系列的论文[4，5，6，7]中，第二位作者给出了Riemann zeta函数的值分布的差异估计，从而得到了Bohr和Bohren经典结果的定量版本[2，3]。本文的目的是加强这些定量结果，同时，从证明中删除对en的超越测度的呼吁（[4]和随后的论文中使用的[9]的结果）。我们还将把[8]的结果改进到它的最佳可能形式（见下面的引理2和下面的注释）。我们首先需要一些符号来说明我们的结果。像往常一样，我们写（（s）为Riemann zeta函数，并输入'<$={s= o+ it：a>\，t# 0，如果（7^\，C（s+ x）# 0 for any JC^ 0}.对于Re 5> 1，定义logC（s）没有困难，并且我们可以通过从2+ it沿着水平线段的解析延拓来定义se $的这个函数。当然，在黎曼假设下，^是整个半平面a>\，|，1]移除。现在设R是复平面中的任意一个闭矩形，其边平行于轴，并将集合的测度记为L（T，R）
In a series of papers [4, 5, 6, 7] the second named author has produced discrepancy estimates for the value-distribution of the Riemann zeta-function, thereby obtaining quantitative versions of Bohr and Jessen's classical results [2, 3]. It is the purpose of this paper to strengthen these quantitative results and, at the same time, to remove from the proofs an appeal to a transcendence measure for en (the result from [9] used in [4] and subsequent papers). We shall also improve the result of [8] to its best possible form (see Lemma 2 below and the Remark following). We first require some notation to state our results. We write, as usual,((s) for the Riemann zeta-function, and put'< $={s= o+ it: a>\, t# 0 if (7^\, C (s+ x)# 0 for any JC^ 0}.For Re 5> 1 there is no difficulty in defining logC (s), and we can define this function for se $ by analytic continuation along the horizontal line segment from 2+ it. Of course, under the Riemann hypothesis^ would be the whole of the half-plane a>\with (|, 1] removed. Now let R be any closed rectangle in the complex plane with edges parallel to the axes, and write L (T, R) for the measure of the set