2007 Fiscal Year Final Research Report Summary
Beneath on analytic properties ofvarious zeta-functions
| Project/Area Number |
17540022
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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 | Nagoya University |
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Principal Investigator |
TANIGAWA Yoshio Nagoya University, Graduate School of Mathematics, Associate Professor (50109261)
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| Co-Investigator(Kenkyū-buntansha) |
KANEMITSU Shigeru Kinki. University, School of Human Oriented Sdence and Engineering, Professor (60117091)
TSUKADA Haruo Kinki University, School of Human Oriented Wend, and Engineering, Associate Professor (00257990)
AKIYAMA Shigeki Niigata University, Graduate School of Science and Technology, Associate Professor (60212445)
KIUCHI Isao Yamaguchi University, Graduate School of Science and Engineeritng, Associate Professor (30271076)
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| Project Period (FY) |
2005 – 2007
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| Keywords | functional equation / modular relation / H-function / Mellin-Barnes integral / double zeta-function / exponential sum / zeta-function associated with polynomial |
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| Research Abstract |
In this research, we succeeded to give a new and the most general formulation of the modular relation by using Meijer's G-function and the Fox H-function as an integral kernel. In this way, we can understand more clearly the role of the functional equations of zeta functions which appeared in many previous works. Our formulation also enables us to generalize many arithmetical formulas in wider context. For example, we generalized Davenport-Segal's arithmetical Fourier series and got interesting examples. We also gave a new proof of the functional equation of the Hurwitz zeta-function and, inspired by the work of Espinosa-Moll and Mikolas, we derived many arithmetically interesting integral formulas. We are now writing a book on our results on the general modular relations.
We are also interested in multiple zeta functions. Especially for double zeta function of Euler-Zagier type, we gave, by employing Krazel's theory of double exponential sum, a non-trivial upper bound in the so-called "critical strip". This is a meaningful improvement of previously known results. We can expect that our result has many applications in the theory of arithmetical functions. We also improved the upper bounds for the triple zeta functions in the critical strip. As another type of zeta functions, we investigated zeta-functions associated with polynomials. We gave the criterion of the possibility of analytic continuation and constructed an example of zeta-function with a natural boundary.
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