2002 Fiscal Year Final Research Report Summary
research on number theoretic concepts attached formal group
| Project/Area Number |
13640016
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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 |
SATOH Junya Nagoya University, Grad.Sch.Human Info., Associate Professor, 大学院・人間情報学研究科, 助教授 (20235352)
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| Co-Investigator(Kenkyū-buntansha) |
TSUKIJI Tatsuie Nagoya University, Grad.Sch.Human Info., Research Associate, 大学院・人間情報学研究科, 助手 (70291961)
YOSHINOBU Yasuo Nagoya University, Grad.Sch.Human Info., Research Associate, 大学院・人間情報学研究科, 助手 (90281063)
YASUMOTO Masahiro Nagoya University, Grad.Sch.Human Info., Professor, 大学院・人間情報学研究科, 教授 (10144114)
MATSUMOTO Kohji Nagoya University, Grad.Sch.Math., Professor, 大学院・多元数理科学研究科, 教授 (60192754)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Number Theory / Formal Group / Zeta Function / Bernoulli Numbers / Distribution Relation |
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| Research Abstract |
(1) We extend a well-known distribution relation for ordinary Bernoulli polynomials to that of Bernoulli polynomials attached to formal group.
(2) Let N be an elementary extension of N and n ∈ N-N. We prove that PTC (n) has no proper endextension of Δ^b_1-LLIND and consider conditions that a model of bounded arithmetic has a proper endextension.
(3) We study the structure of uniform random binary recursive circuits. We show that a suitably normalized version of the number of outputs converges in distribution to a normal random variate. We also discuss the connection of the number of outputs to a non-classical urn model, and our investigation provides a first solved instance of this new class of urns.
(4) Let ζ(s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form Σ^q_<a=1> ζ(s, a/q)^h or Σ^q_<a=1> |ζ(s, a/q)|^<2h> are studied, where q and h are positive integers. These mean values can be written as linear combinations of Σ^q_<a=1> ζ_r(s1, ・・・, sr, a/q), where ζ_r(s1, ・・・, sr;α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of Σ^q_<a=1> ζ_r(s1, ・・・, sr;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for Σ^q_<a=1> ζ(s, a/q)h and Σ^q_<a=1> |ζ(s, a/q)|^<2h> is obtained. In particular, the asymptotic expansion of Σ^q_<a=1> ζ(1/2, a/q)^3 with respect to q is written down.
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