10.46298/hrj.1980.88
https://hrj.episciences.org/88
Ramachandra, K
K
Ramachandra
Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II
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 function-theoretic 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 half-plane $\sigma>1$.
episciences.org
Dirichlet series
mean value theorems
Riemann zeta-function
[MATH] Mathematics [math]
2015-06-12
1980-01-01
1980-01-01
en
journal article
https://hal.archives-ouvertes.fr/hal-01103855v1
2804-7370
https://hrj.episciences.org/88/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 3 - 1980
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