| Project/Area Number |
10640029
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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 | Yamaguchi University |
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Principal Investigator |
KIUCHI Isao Faculty of Science, Yamaguchi University, Associate Professor, 理学部, 助教授 (30271076)
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| Co-Investigator(Kenkyū-buntansha) |
TANIGAWA Yoshio Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (50109261)
YANAGIHARA Hiroshi Faculty of Engineering, Yamaguchi University, Associate Professor, 工学部, 助教授 (30200538)
MASUMOTO Makoto Faculty of Science, Yamaguchi University, Associate Professor, 理学部, 助教授 (50173761)
KIKUMASA Isao Faculty of Science, Yamaguchi University, Associate Professor, 理学部, 助教授 (70234200)
MATSUMOTO Kohji Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (60192754)
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| Project Period (FY) |
1998 – 1999
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| Project Status |
Completed (Fiscal Year 1999)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1998: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Riemann zeta-function / Divisor function / Riemann-Siegel formula / Omega results / Multiplicative functions / Mean value theorems / 平均値定理 / 平均値公式 / short intervals / 数論的関数 / Ω-結果 / Voronoiの公式 |
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| Research Abstract |
This investigations are on the mean value theorems of Riemann's zeta-function, and an exponential sum involving the generalized divisor function. We study it by using the methods of Jutila, the approximate functional equation of Motohashi, the Atkinson formula of Matsumoto-Meurman and the Riemann-Siegel formula, introduced by M. Jutila, M. Motohashi, K. Matsumoto and T. Meurman in their consideration for the theory of zeta-function. Particular we study mean square of the remainder term for their summatory functions. The main subjects treated here are the following five : (1)the divisor problem for short intervals, (2)the Matsumoto-Meurman formula for short intervals, (3)mean square for the non-symmetric form of the approximate functional equation of Motohashi, (4)mean square for the non-symmetric form of the approximate functional equation of Motohashi for short intervals, (5)mean square for the Riemann-Siegel formula. For each subject, we have obtained the following results and forekn
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owledges :
(1), (2)The mean square of the remainder term for summatory of the divisor function for short intervals was first by M. Jutila, who obtained the asymptotic formula involving the integral for short intervals. The result of this note is the mean value formula of the remainder term for an exponential sum involving the generalized divisor functions for short intervals. This result is an analoge of Jutila's result. Similarly, by using Jutila methods, we are derived to the mean value theorem of the remainder term for the mean value formula of the Riemann zeta-function in the critical strip.
(3), (4)A very estimation for remainder term of the approximate functional equation for the square of Riemann zeta-function was first obtained by Hardy-Littlewood in 1929, but Motohashi improved it to an analogue of the Riemann-Siegel formula for the square of the Riemann zeta-function in 1983. The results of this note are derived to the mean value formula for remainder term, and the mean value theorem of this remainder term for short intervals.
(5)As application of the Riemann-Siegel formula, we have the even power moments for the remainder term of this formula. Less
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