摘要
1859年,Riemann以Euler恒等式作为研究的出发点,定义了复变数s=σ+it的函数—Riemann Zeta函数,对Zeta函数进行了非常深刻的研究,解析数论也正是沿着Riemann所指明的方向在二十世纪取得了迅速的发展.Riemann Zeta函数的零点与素数的分布有着非常密切的关系.首先简述了Riemann Zeta函数的解析性质:函数方程、非零区域、阶的估计、积分均值等,对Riemann Zeta函数的零点分布的研究动态进行了阐述,并利用零点密度估计的经典方法—零点探测法,证明了Ingham的经典结果.最后介绍了Riemann Zeta函数的高阶推广—自守L-函数的零点分布及应用的研究进展,其中也包括了作者近年来在这一领域所做的部分工作.
In 1859,Riemann defined the function of complex variable s=σ+it-Riemann Zeta function from the Euler′s identity,and made a very deep research on Riemann Zeta function.Analytic number theory developed rapidly in the 20th century just along the direction pointed out by Riemann.The zeros of Riemann Zeta function are closely related to the distribution of prime numbers.In this paper,analytic properties of Riemann Zeta function,such as functional equation,zero-free region and order,are briefly introduced.Also we use the classical method zero detection method to derive Ingham′s result.At last,we introduce the research progresses of the zero distribution and its applications for Riemann Zeta function and automorphic L-function,which is the generalization of Riemann Zeta function,some work done by the author in this field in recent years are included.
作者
张德瑜
黄敬
张芮
Zhang Deyu;Huang Jing;Zhang Rui(School of Mathematics and Statistics, Shandong Normal University, 250358, Jinan, China)
出处
《山东师范大学学报（自然科学版）》
CAS
2020年第1期22-30,共9页
Journal of Shandong Normal University(Natural Science)
基金
国家自然科学基金资助项目(11771256)
山东省自然科学基金资助项目(ZR2015AM010).
