Estimate of the sum of arithmetical functions and its applications to L functions

算术函数之和的估计及其在 L 函数中的应用

• 批准号: 09640026
• 负责人: TANIGAWA Yoshio
• 金额: $ 1.86万
• 依托单位: Nagoya University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640026/
• 关键词: divisor problem; Voronoi formula; Rankin-Selberg series; mean value formula; elliptic curve; Riemann hypothesis; Voronoi 級数; ゼータ関数

In this research, we studied the sum of various arithmetical functions, and got the estimate of remainder term, the mean square formula and OMEGA-results. 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 L-function associated to elliptic curves. We found the effective methods for numerical computation of L-function 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 Sato-Tate conjecture and the Riemann hypot

在这项研究中，我们研究了各种算术函数的总和，并获得了剩余术语，均方根公式和欧米茄的估计。首先，我们处理了兰金问题，即模块化形式的傅立叶系数的平方之和。 1939年，兰金通过Landau定理获得了剩余期限的主术语和上限。我们从Voronoi公式的角度研究了这个问题。我们获得了剩余期限与第一个riesz平均值之间的密切关系。例如，第一个Riesz平均值的更精确的正方形平均公式使我们改善了其余项。我们可以期望进一步的分析，因为第一个Riesz平均值的Voronoi公式是收敛的。对于局部行为的研究，短时间间隔的均方根公式很有用。我们以简短的间隔获得了概括的除数函数的公式。我们还获得了Xi（s） ^2平均值的误差项的类似估计。这些结果是Jutila结果的类似物。我们还考虑了与椭圆曲线相关的L功能。我们发现了L功能数值计算的有效方法，并检查了Riemann Htpothesiss在几个椭圆曲线的IM（S）<小于或等于> 400的范围内。此外，我们研究了Sato-Tate猜想与Riemann Hypot之间的关系

项目成果

S.Akiyama: "Almost uniform distribution modulo 1 and the distribution of primes" Acta Math.Hungar.78. 39-44 (1998)

S.Akiyama：“模 1 几乎均匀分布和素数分布”Acta Math.Hungar.78。

S.Akiyama: "Pisot numbers and greedy algorithm" “Number Theory,Diophantine,Computational and Algebraic Aspects" (ed.by K.Gyory,A.Petto,V.T.Sos). 9-21 (1998)

S.Akiyama：“皮索数和贪婪算法”“数论、丢番图、计算和代数方面”（K.Gyory、A.Petto、V.T.Sos 编辑）9-21 (1998)。

A.Ivic, K.Matsumoto and Y.Tanigawa: "On Riesz means of the coeffcients of the Rankin-Selberg series" Math.Proc.Camb.Phil.Soc.(to appear).

A.Ivic、K.Matsumoto 和 Y.Tanikawa：“On Riesz 表示 Rankin-Selberg 级数的系数”Math.Proc.Camb.Phil.Soc.（即将出现）。

M.Katsurada and K.Matsumoto: "Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions.II" NEW TRENDS IN PROBABILITY AND STATISTICS (Analytic and Probabilistic Methods in Number Theory). 4. 119-134 (1997)

M.Katsurada 和 K.Matsumoto：“Hurwitz zeta 函数的某些均方的显式公式和渐近展开。II”概率和统计的新趋势（数论中的分析和概率方法）。

I.Kiuchi and Y.Tanigawa: "The mean value theorem of the Riemann zeta-function in the critical strip for short intervals" “Number Theory and its Applications",(ed.by K.Gyory and S.Kanemitsu). to appear.

I.Kiuchi 和 Y.Tanikawa：“短区间临界带中黎曼 zeta 函数的均值定理”“数论及其应用”，（K.Gyory 和 S.Kanemitsu 编辑）。 。