Project/Area Number  09640026 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  Nagoya University 
Principal Investigator 
TANIGAWA Yoshio Nagoya Univ.Graduate School of Mathematics, Assoc.Prof., 大学院・多元数理科学研究科, 助教授 (50109261)

CoInvestigator(Kenkyūbuntansha) 
SUZUKI Hiroshi Nagoya Univ.Graduate School of Mathematics, Lecturer, 大学院・多元数理科学研究科, 講師 (70235993)
AKIYAMA Shigeki Nagoya Univ.Faculty of Science, Assoc.Prof., 理学部, 助教授 (60212445)
MATSUMOTO Kohji Nagoya Univ.Graduate School of Mathematics, Assoc.Prof., 大学院・多元数理科学研究科, 助教授 (60192754)
KITAOKA Yoshiyuki Nagoya Univ.Graduate School of Mathematics, Prof., 大学院・多元数理科学研究科, 教授 (40022686)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,900,000 (Direct Cost : ¥2,900,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1997 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  divisor problem / Voronoi formula / RankinSelberg series / mean value formula / elliptic curve / Riemann hypothesis / 約数問題 / Voronoi公式 / 2乗平均 / 楕円曲線 / Riemann予想 / Voronoi 級数 / ゼータ関数 
Research Abstract 
In this research, we studied the sum of various arithmetical functions, and got the estimate of remainder term, the mean square formula and OMEGAresults. First we treated the Rankin problem, namely the sum of square of Fourier coefficients of modular forms. In 1939, Rankin obtained, via Landau theorem, the main term and the upper bound of the remainder term. We studied this problem from a point of Voronoi formula. We obtained the close relationship between the remainder term and the first Riesz mean of it. For example, more precise square mean formula of the first Riesz mean gives us the improvement of the remainder term. We can expect further analysis because the Voronoi formula of the first Riesz mean is convergent. For the study of local behaviour, the mean square formula for short interval is useful. We got the formula in short interval for generalized divisor function. We also obtainded the similar estimate for the error term of the mean of xi (s) ^2. These results are analogoue of Jutila's results. We also considered the Lfunction associated to elliptic curves. We found the effective methods for numerical computation of Lfunction and checked that the Riemann htpothesis holds in the range Im (s) <less than or equal> 400 for several elliptic curves. Furthermore, we studied the relation between SatoTate conjecture and the Riemann hypot
