Shigeru Kanemitsu

Dirichlet series with periodic coefficients

In this paper we shall unify the results obtained so far in various scattered literature, for Dirichlet characters and the associated Dirichlet L-functions, under the paradigm of periodic arithmetic functions and the associated Dirichlet series. Notably we shall determine the Laurent coefficients of the series in question to cover Funakura’s result and proceed on to prove the Ayoub-Berndt-Carlitz-Chowla-Müller-Redmond theorem.

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.

On the Hurwitz—Lerch zeta-function

Summary. Let

Farey series and the Riemann hypothesis

\Phi(z,s,\alpha) = \sum\limits^\infty_{n = 0} {z^n \over (n + \alpha)^s}

Ramanujan’s Formula and Modular Forms

be the Hurwitz-Lerch zeta-function and

On Rapidly Convergent Series Expressions for Zeta- and L-Values, and Log Sine Integrals

\phi(\xi,s,\alpha)=\Phi(e^{2\pi i\xi},s,\alpha)

ON THE VALUES OF A CLASS OF DIRICHLET SERIES AT RATIONAL ARGUMENTS

for

On Bessel series expressions for some lattice sums: II

\xi\in{\Bbb R}

Weighted short-interval character sums

its uniformization.

SOME NUMBER THEORETIC APPLICATIONS OF A GENERAL MODULAR RELATION

\Phi(z,s,\alpha)