关于ζ函数在点的组合处商的ω定理。

omega-Theorems for Quotients of Zeta-Functions at Combinations of Points.

作者信息

Levinson N

机构信息

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass. 02139.

出版信息

Proc Natl Acad Sci U S A. 1972 Sep;69(9):2528-9. doi: 10.1073/pnas.69.9.2528.

DOI:10.1073/pnas.69.9.2528

PMID:16592011

原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC426980/

Abstract

Theorem A. Let q >/= O and r >/= O be integers. Let s = sigma + it, let zeta(s) be the Riemann zeta-function, let G(o)(s) = 1, and [Formula: see text] and let F(s) = G(q)(s)/H(r)(s). Then as t --> infinity lim sup [unk]F(1 + it)[unk]/(log log t)(q+r+1) >/= (6/pi)(2))(r+1) exp {(q + r + 1)gamma}, where gamma is Euler's constant.Stronger results such as proved in [1] are valid, and in particular q and r can be allowed to increase with t as in [1]. Results involving the real part of the sum of the factors of G(q) and of the reciprocals of the factors of H(r) can be proved much as in [3].

摘要

定理A。设q≥0且r≥0为整数。设s = σ + it，ζ(s)为黎曼ζ函数，设G(0)(s) = 1，且[公式：见文本]，并设F(s) = G(q)(s)/H(r)(s)。那么当t→∞时，lim sup |F(1 + it)|/(log log t)(q + r + 1)≥(6/π)(2)(r + 1) exp{(q + r + 1)γ}，其中γ为欧拉常数。[1]中证明的更强结果是有效的，特别地，q和r可以像[1]中那样随t增大。涉及G(q)的因子之和的实部以及H(r)的因子的倒数之和的实部的结果，证明方式与[3]中大致相同。