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2021 Fiscal Year Final Research Report

Study of error terms in analytic number theory

Research Project

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Project/Area Number 19K03449
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionYamaguchi University

Principal Investigator

Minamide Makoto  山口大学, 大学院創成科学研究科, 准教授 (80596552)

Project Period (FY) 2019-04-01 – 2022-03-31
Keywords解析数論的誤差項 / 二乗平均 / 様々な約数関数 / メビウス関数 / リーマンゼータ関数 / ハーディー関数 / 微分 / 二重ゼータ関数
Outline of Final Research Achievements

In 1999, Hall studied the mean square of the derivatives of Hardy's Z-function, and suggested to improve the result. It was a main purpose of our project. Yoshio Tanigawa and I succeeded to show the O-estimate as Hall' suggestion. On double zeta function, an estimate of the order of function was suggested by Isao Kiuchi and Y. Tanigawa, 2006. Debika Banerjee, Y. Tanigawa and I attempted to prove the conjecture and we obtained it. Moreover, we improved a result on the mean square of double zeta function by I. Kiuchi and I. Joshi and Vaidya studied the error term in the mean of the greatest divisor of n which is coprime to fixed k. Jun Furuya, Miyu Nakano, and I considered the error term in the mean square of the function. In the case of k=p (any prime), we obtained the limsup and liminf of the error term. As a related problem, for any square-free integer k, Tadaaki Igawa, M. Nakano and I obtained an asymptotic formula of the mean of the error term.

Free Research Field

解析的整数論

Academic Significance and Societal Importance of the Research Achievements

およそ20年, 多重ゼータ関数の研究は日本を中心として発展し続けている. この様な状況において, 当初予定になかった, 二重ゼータ関数の評価や二乗平均について, 知られている結果の改良を得られたことは学術的にも社会的にも意義があったと思う. また, Hall のハーディー関数の微分の二乗平均の結果を改良できたことは, リーマンゼータ関数の理論に寄与したと思う. 他に, インドで盛んに研究されていたある種の約数関数についての研究について, 今回, 二乗平均に関する結果が得られたことは意義があると思う.

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Published: 2023-01-30  

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