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

本文研究了各种数论函数的和，得到了余项的估计、均方公式和Ω结果。首先，我们处理的兰金问题，即平方和的傅立叶系数的模形式。1939年，兰金利用朗道定理得到了主项和余项的上界。本文从Voronoi公式的角度研究了这一问题。我们得到了余项与其第一Riesz平均之间的密切关系，例如，更精确的第一Riesz平均的平方平均公式给了余项的改进。由于第一Riesz平均的Voronoi公式是收敛的，我们可以期待进一步的分析。对于局部行为的研究，短区间的均方公式是有用的。给出了广义除子函数在短区间内的计算公式。对于Xi（s）^2的均值误差项，我们也得到了类似的估计.这些结果类似于Jutila的结果。我们还考虑了与椭圆曲线相关的L-函数。我们找到了数值计算L-函数的有效方法，并对几种椭圆曲线检验了Riemann猜想在Im（s）400范围内成立<less than or equal>。进一步研究了Sato-Tate猜想与Riemann hypot的关系

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