episciences.org_88_1653621205
1653621205
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
01
01
1980
Volume 3  1980
Some remarks on the mean value of the riemann zetafunction and other Dirichlet seriesII
K
Ramachandra
This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{C}$ represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate $2k$ is an integer.
%This is of great interest, for little has been known on the mean value of $\vert\zeta(\frac{1}{2}+it)\vert^k$ for odd $k$, say $k=1$; for even $k$, see the book by E. C. Titchmarsh [The theory of the Riemann zeta function, Clarendon Press, Oxford, 1951, Theorem 7.19].
The proofs are based on applications of classical functiontheoretic theorems, together with mean value theorems for Dirichlet polynomials or series.
%In the case of the zeta function, the principle is to write $\vert\zeta(s)\vert^k=\vert\zeta(s)^{k/2}\vert^2$, where $\zeta(s)^{k/2}$ is related to a rapidly convergent series which is essentially a partial sum of the Dirichlet series of $\zeta(s)^{k/2}$, convergent in the halfplane $\sigma>1$.
01
01
1980
88
https://hal.archivesouvertes.fr/hal01103855v1
10.46298/hrj.1980.88
https://hrj.episciences.org/88

https://hrj.episciences.org/88/pdf