{"docId":88,"paperId":88,"url":"https:\/\/hrj.episciences.org\/88","doi":"10.46298\/hrj.1980.88","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":14,"name":"Volume 3 - 1980"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01103855","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01103855v1","dateSubmitted":"2015-03-03 16:13:34","dateAccepted":"2015-06-12 16:05:11","datePublished":"1980-01-01 08:00:00","titles":{"en":"Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II "},"authors":["Ramachandra, K"],"abstracts":{"en":"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":[{"en":"Dirichlet series"},{"en":"mean value theorems"},{"en":"Riemann zeta-function"},"[MATH] Mathematics [math]"]}