Hardy-Ramanujan Journal |
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 (loglogH)−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 |ζ(12+it)|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 |ζ(s)|k=|ζ(s)k/2|2, where ζ(s)k/2 is related to a rapidly convergent series which is essentially a partial sum of the Dirichlet series of ζ(s)k/2, convergent in the half-plane σ>1.