2004 Fiscal Year Final Research Report Summary
Zeta-functions and hypergeometric functions
| Project/Area Number |
14540051
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kinki University |
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Principal Investigator |
KANEMITSU Shigeru Kinki University, Faculty of Human-oriented Science and Technology, Department of Information Science, Professor, 産業理工学部, 教授 (60117091)
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| Co-Investigator(Kenkyū-buntansha) |
TANIGAWA Yoshio Nagoya University, Graduate School of Polymathematics, Associate Professor, 大学院, 助教授 (50109261)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Zeta-functions / Modular relation / Bessel series expansion / Crystal / Madeling constant / Hurwitz zeta-function / Screened Coulomb potential / Epstein zeta-function |
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| Research Abstract |
The expected objective of studying the functional equation through the modular relation was almost completed thanks to the generous support of the research grant from the JSPS. Now we are ready to launch on the next stage of covering the theory of zeta-functions which has been studied in the last 160 years from the time of Eisenstein and Riemann.
In this year we applied our results that we obtained in the previous two years to related fields of physics and chemistry. Namely we studied the fundamental subject of these disciplines-crystal (structures) and the associated constant, the Madelung constant-numerically through the associated Epstein zeta-function In three of the papers that were published in 2004, we have succeeded in incorporating all the existing results into the framework of modular relations, or the Bessel series expansion or the incomplete gamma series as their manifestations.
Also in the mean square theory of zeta-and L-functions, we have extracted the core of the stuff, the Euler digamma function, and succeeded in deriving the complete asymptotic expansion for the means square of the value of the Dirichlet L-function at 1, by making full use of the special function-theoretic aspects, thus making it possible to push the situation forward further to cover the case of the Lerch zeta-function.
As regards the Hurwitz zeta-function, we completed our research on the derivation of all information on it from its partial sum and now we are able to derive all formulas needed e.g. in zeta-regularization.
Finally, we organized an international symposium on "Zeta functions, Topology and Quantum Physics" at Kinki University, Osaka in 2003 and as its continuation we organized another one in 2004 about "Quantum Computation". We are going to publish the proceedings of these two symposia.
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