episciences.org_159_1653170507
1653170507
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
2007
Volume 30  2007
Contributions to the theory of the Hurwitz zetafunction
S
Kanemitsu
Y
Tanigawa
H
Tsukada
M
Yoshimoto
We give various contributions to the theory of Hurwitz zetafunction. 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 farreaching 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 zetafunction 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 EspisonaMoll results.
%In this paper we shall give various contributions to the theory of the Hurwitz zetafunction. 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 psifunction, 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.
01
01
2007
159
https://hal.archivesouvertes.fr/hal01112081v1
10.46298/hrj.2007.159
https://hrj.episciences.org/159

https://hrj.episciences.org/159/pdf