HAL will be down for maintenance from Friday, June 10 at 4pm through Monday, June 13 at 9am. More information

Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II

Abstract : 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 $\sigma>1$.
Keywords :
Document type :
Journal articles
Domain :

Cited literature [6 references]

https://hal.archives-ouvertes.fr/hal-01103855
Contributor : Ariane Rolland Connect in order to contact the contributor
Submitted on : Thursday, January 15, 2015 - 3:31:21 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
Long-term archiving on: : Thursday, April 16, 2015 - 10:51:09 AM

File

3Article1.pdf
Explicit agreement for this submission

Citation

K Ramachandra. Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1980, Volume 3 - 1980, pp.1 - 24. ⟨10.46298/hrj.1980.88⟩. ⟨hal-01103855⟩

Record views