Mean values and Asymptotic Behavior on Arithmetical Functions

算术函数的平均值和渐近行为

基本信息

批准号：
10640029
负责人：
KIUCHI Isao
金额：
$ 2.11万
依托单位：
Yamaguchi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640029/
关键词：
Riemann zeta-function Divisor function Riemann-Siegel formula Omega results Multiplicative functions Mean value theorems 平均値定理平均値公式 short intervals 数論的関数 Ω-結果 Voronoiの公式

项目摘要

This investigations are on the mean value theorems of Riemann's zeta-function, and an exponential sum involving the generalized divisor function. We study it by using the methods of Jutila, the approximate functional equation of Motohashi, the Atkinson formula of Matsumoto-Meurman and the Riemann-Siegel formula, introduced by M. Jutila, M. Motohashi, K. Matsumoto and T. Meurman in their consideration for the theory of zeta-function. Particular we study mean square of the remainder term for their summatory functions. The main subjects treated here are the following five : (1)the divisor problem for short intervals, (2)the Matsumoto-Meurman formula for short intervals, (3)mean square for the non-symmetric form of the approximate functional equation of Motohashi, (4)mean square for the non-symmetric form of the approximate functional equation of Motohashi for short intervals, (5)mean square for the Riemann-Siegel formula. For each subject, we have obtained the following results and forekn … More owledges :(1), (2)The mean square of the remainder term for summatory of the divisor function for short intervals was first by M. Jutila, who obtained the asymptotic formula involving the integral for short intervals. The result of this note is the mean value formula of the remainder term for an exponential sum involving the generalized divisor functions for short intervals. This result is an analoge of Jutila's result. Similarly, by using Jutila methods, we are derived to the mean value theorem of the remainder term for the mean value formula of the Riemann zeta-function in the critical strip.(3), (4)A very estimation for remainder term of the approximate functional equation for the square of Riemann zeta-function was first obtained by Hardy-Littlewood in 1929, but Motohashi improved it to an analogue of the Riemann-Siegel formula for the square of the Riemann zeta-function in 1983. The results of this note are derived to the mean value formula for remainder term, and the mean value theorem of this remainder term for short intervals.(5)As application of the Riemann-Siegel formula, we have the even power moments for the remainder term of this formula. Less

这项研究是关于里曼Zeta功能的意义价值理论，以及涉及广义分裂函数的指数总和。我们使用Jutila的方法，Motohashi的近似功能方程，Matsumoto-Meurman的Atkinson公式和Riemann-Siegel Formula，由M. Jutila，M。Motohashi，M。Motohashi，K。Matsumoto和T. Meurman介绍了Zeta-funuttion的考虑。特别是我们研究其余术语的均值平方。此处处理的主要受试者是以下五个：（1）短间隔的除数问题，（2）短间隔的均舒人 - 穆尔曼公式，（3）（3）Motohashi近似功能方程的非对称形式的均值平方平方平方，（4）（4）MotohaShi近似平方的均值平方的均值（4）Motohashi均值（5）的平方平方（5）（5）公式。对于每个受试者，我们都获得了以下结果和前景…更多的猫头鹰：（1），（2）Jutila M. Jutila首先首先是剩余的分隔函数的余额，用于简短间隔，后者获得了涉及短间隔的积分的不对称公式。本说明的结果是涉及短时间间隔的广义分数函数的剩余项的平均值公式。这个结果是朱迪拉结果的类似物。 Similarly, by using Jutila methods, we are derived to the mean value theory of the remainder term for the mean value formula of the Riemann zeta-function in the critical strip.(3), (4)A very estimation for remainder term of the approximate functional equation for the square of Riemann zeta-function was first obtained by Hardy-Littlewood in 1929, but Motohashi improved it to an analogue of the Riemann-siegel公式为1983年Riemann Zeta功能的平方。该注释的结果得出余量术语的平均值公式，以及该余下项的平均值理论。（5）作为Riemann-Siegel公式的应用，我们具有该剩余公式的均匀力量。较少的