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和，并得到了余项的估计、均方公式和OMEGA结果。首先，我们处理兰金问题，即模形式的傅里叶系数的平方和。 1939年，兰金通过朗道定理得到了主项和余项的上界。我们从Voronoi公式的角度研究了这个问题。我们得到了余项与其一阶Riesz均值之间的密切关系。例如，第一个 Riesz 均值的更精确的均方公式为我们提供了余项的改进。我们可以期待进一步的分析，因为第一 Riesz 均值的 Voronoi 公式是收敛的。对于局部行为的研究，短间隔的均方公式很有用。我们得到了广义除数函数的短区间公式。我们还获得了 xi (s) ^2 均值误差项的类似估计。这些结果与 Jutila 的结果类似。我们还考虑了与椭圆曲线相关的 L 函数。我们找到了 L 函数数值计算的有效方法，并检查了对于几个椭圆曲线，黎曼假说在 Im (s) <小于或等于> 400 的范围内成立。此外，我们还研究了佐藤泰特猜想与黎曼假设之间的关系

项目成果

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）。 。

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）。

J.Furuya and Y.Tanigawa: "Estimation of a certain function related to the Dirichlet divisor problem" “Analytic and Probabilistic Methods in Number Theory",New Trends in Probability and Statistics (ed.by A.Laurincikas,et al.). 4. 171-189 (1997)

J.Furuya 和 Y.Tanikawa：“与狄利克雷除数问题相关的某个函数的估计”“数论中的分析和概率方法”，概率与统计新趋势（A.Laurincikas 等人编辑）。 4. 171-189（1997）

S.Akiyama: "Self affine tiling and Pisot numeration system" "Number Theory and its Applications", (ed by K.Gyory and S.Kanemitsu). to appear.

S.Akiyama：“自仿射平铺和皮索计数系统”“数论及其应用”，（由 K.Gyory 和 S.Kanemitsu 编辑）。