Hardy Ramanujan Journal |

We give various contributions to the theory of Hurwitz zeta-function. An elementary part is the argument relating to the partial sum of the defining Dirichlet series for it; how much can we retrieve the whole from the part. We also give the sixth proof of the far-reaching Ramanujan -- Yoshimoto formula, which is a closed form for the important sum $\sum^\infty_{m=2} \frac{\zeta(m,\alpha)}{m+\lambda} z^{m+\lambda}$. This proof, incorporating the structure of the Hurwitz zeta-function as the principal solution of the difference equation, seems one of the most natural ones. The formula may be applied to deduce almost all formulas in H.~M.~Srivastava and J.~Choi. The same is applied to obtain closed form for the integral of the Euler psi function and give Espisona-Moll results. %In this paper we shall give various contributions to the theory of the Hurwitz zeta-function. In \S1 we shall continue our previous study and give integral representations (for the derivatives as well) which give another basis of the theory of gamma and related functions. In \S2 we shall give the sixth proof of the Ramanujan formula with two examples which supersede those results presented in the book of Srivastava and Choi. In \S3 we shall give two more proofs of the closed formula for the integral of the psi-function, thus recovering the recent result of Episona and Moll. Finally, in \S4 we shall give another proof of the functional equation. Hereby we put all existing literature in the hierarchical and historical perspective.

Source : oai:HAL:hal-01112081v1

Volume: Volume 30

Published on: January 1, 2007

Submitted on: March 3, 2015

Keywords: Ramanujan-Yoshimoto formula, partial sum, Espisona-Moll formulas, integral of the psi function, series involving zeta-functions, difference equation,Hurwitz zeta-function,[MATH] Mathematics [math]

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