Sum formulas for arithmetical functions and mean value theorem for the zeta functions

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The Riemann zeta-function, the Dirichlet L-fucntion and other Dirichlet series have a long history of researches. They are still the main oobjects of researches at present day, since many important number theoretic properties are reflected in the analytic properties of Dirichlet series, such as the analtic continuation, functional equation, the place of poles and their residues.To study the detailed local behavior of zeta function, Kiuchi and Tanigawa derived mean value theorem for short intervals for E_σ (T)(=the error term of the mean square formula of ζ^2 (σ+it) and R (σ+it)(=the remainder term of the approximate functional equation of ζ^2 (s)). We also studied the mean value of eror term arising from the Dirichlet divisor problem with characters.Akiyama and Tanigawa considered the L-function associated to elliptic curves. In order to compute the numerical values, we derived the approximate functional equation with incomplete gamma functions. Using our formula, we checked that the Riemann hypothesis holds in the range Im (s) 【less than or equal】400 for several elliptic curves. Furthermore, we studied the relation between Sate-Tate conjecture and the Riemann hypothesis.The above mentioned functional equation is related to the modular relation. Kanemitsu, Yoshimoto and Tanigawa reconstruct many results related to the Ramanujan formulas from the point of modular relations. The new formula for ζ(2/3) of Ramanujan type is also obtained.The analytic continuation of Euler-Zagier's multiple zeta-function was established by Akiyama, Egami and Tanigawa. We noted for the first time that the points of indeterminacy appear for this function. We also computes the zeta-values at non-positive integers.