2007 Fiscal Year Final Research Report Summary
Zeta-functions and special functions
| Project/Area Number |
17540050
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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, School of Humanity-Oriented Science and Engineering, Professor (60117091)
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| Co-Investigator(Kenkyū-buntansha) |
TSUKADA Haruo Kinki University, School of Humanity-Oriented Science and Engineering, Asso. Professor (00257990)
TANIGAWA Yoshio Kinki University, Garduate School of Polymathematics, Asso. Professor (50109261)
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| Project Period (FY) |
2005 – 2007
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| Keywords | zeta-functions / functional equation / modular relation / Fox H-function / Riemann hypothesis |
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| Research Abstract |
In these three years, I have been working on the equivalent forms of the functional equations for the zeta-functions. We have made it clear that the essential ingredient in the equivalent forms is the Fox H-functions and we have succeeded in formulating the most general form of the modular relation which is expected to be applied to various fields where the zeta-functions appear. We have also revealed that many of the existing relations that look independent of the functional equation are indeed, disguised form thereof typically, the partial fraction expansion for the cotangent function is one.
In the cited papers which are published in 2006, we have presented various aspects of the modular relation. Especially, in aspects we have interpreted the functional equation of the Hurwitz-Lerch zeta-function as a manifestation of the modular relation. In applications we have made an essential use of the Fox H-functions to give all possible equivalent assertions to the functional equation (with
… More
gamma factor appearing on one side only) in the form of the modular relation.
In 2007 I have published one book and one proceedings volume. The book is coauthored by the joint researcher Dr. Tsukada, and we state it as an achievement. The contents are as follows. Through the theory of zeta-functions, we may develop the new construction of the theory of special functions including Bernoulli polynomials, gamma function. It also contains the theory of Epstein zeta-functions and its applications to ionic crystals as well as the theory of the Dirichlet L-functions and its applications to class number formulas. The second is the proceedings of the 4?th China-Japan Seminar on number theory-Sailing on the sea of number theory, which contains far-reaching survey papers ranging from analytic number theory to algebraic number theory and is intended for breeding the new generations in both Japan and China. This happens to be Vol. 2 of the book series “Number Theory and its Applications" which the reporter has been editing. In 2007 Vol. 4, by M. Hata has appeared, “Problems in analysis" which contains worked-out problems in analysis and related fields.
In 2007 there appeared two papers. One is about the Hurwitz zeta-function in which we give a proof based on the difference equation satisfied by it to prove Ramanujan's formula which in turn can compress 215 pages of the book by Srivastava and Choi, and as a whole constitutes the culmination of the studies in this direction. Less
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Research Products
(11 results)
