The mean value theorem of an arithmetical error term in short intervals and its application

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04778
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Tanigawa Yoshio 名古屋大学, 多元数理科学研究科, 招へい教員 (50109261)
• Co-Investigator(Renkei-kenkyūsha): MINAMIDE Makoto 山口大学, 大学院創成科学研究科, 講師 (80596552)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 短区間平均値定理 / 数論的誤差項 / Tong型の公式 / 3 次元約数問題 / 符号変化 / shifted convolution / リーマンゼータ関数の導関数 / 近似関数等式

Outline of Final Research Achievements

We derived the mean square formula in short intervals for an arithmetical error term by using the Tong-type truncated formula which we had already established for zeta-functions in the extended Selberg class. We applied our theorems to the study of the three-dimensional divisor problems and got the new results for the shifted convolutions and the sign changes of its arithmetical error term. Furthermore,we remade our Tong-type truncated formula in term of the exponent of the mean square of the zeta-function on the critical line.
In 1999, Hall derived the approximate functional equations for the square of the derivative of the Reimann zeta-function and other functions. By generalizing the method of Titchmarsh, we improved the estimates of Hall's error terms.

Free Research Field

解析的整数論