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

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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