2021 Fiscal Year Final Research Report
Multiple and weighted averaging of zeta and theta functions--their formulations and asymptotics--
| Project/Area Number |
17K05182
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Keio University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | zeta-function / theta function / asymptotic expansion / mean value |
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| Outline of Final Research Achievements |
i) Averaging of the values of Lerch zeta-functions: The head investigator has shown that complete asymptotic expansions exist for the Laplace-Mellin and Riemann-Liouville transforms, together with their appropriate iterations, of Lerch zeta-functions in terms of their pivotal variable $s$ of the transforms, when $s\to0$ and $s\to\infty$ both through the sector $|\arg s|<\pi$. The region of validity of these asymptotic expansions contain any vertical ray through imaginary directions; this allows us in general fairly nice applicability to the problems of analytic number theory;
ii) Asymptotic expansions associated with Dirichlet-Hurwitz-Lerch holomorphic Eisenstein series: The head investigator, jointed with (his collaborator) Professor Takumi Noda, have established complete asymptotic expansions exist for Dirichlet-Hurwitz-Lerch holomorphic Eisenstein series when the associated parameter $z$ of the series tends to $0$ and $\infty$ both through the complex upper half-plane $0<\arg z<\pi$.
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| Free Research Field |
解析的整数論
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| Academic Significance and Societal Importance of the Research Achievements |
i) Lerch ゼータ関数の平均化:ゼータ関数に対する種々の積分変換を考察する研究は，これまでは Laplace 変換や Mellin 変換に関するものが主流であったが，今回，本研究で得られた成果から，新たに Laplace-Mellin 型, Riemann-Liouville 型や，それらの適切な iterations(s) 等の新たなクラスに対しても意義ある結果を導出できることが判明した；
ii) Dirichlet-Hurwitz-Lerch 正則 Eisenstein 級数に付随する漸近展開：表記の漸近展開から，Ramanujan による著名な公式等を含む極めて広範な応用も得られる．
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