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

hrj:88 - Hardy-Ramanujan Journal, January 1, 1980, Volume 3 - 1980 - https://doi.org/10.46298/hrj.1980.88
Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II

Authors: 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 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 $\sigma>1$.


Volume: Volume 3 - 1980
Published on: January 1, 1980
Imported on: March 3, 2015
Keywords: Dirichlet series,mean value theorems,Riemann zeta-function,[MATH] Mathematics [math]


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