Abstract

The “hybrid” moments ∫T2Tζ1/2+itk∫t-Gt+Gζ1/2+ixldxmdt Tε≪G=GT≪T of the Riemann zeta-function ζs on the critical line Res=1/2 are studied. The expected upper bound for the above expression is Oε(T1+εGm). This is shown to be true for certain specific values of k,l,m∈N, and the explicitly determined range of G=G(T;k,l,m). The application to a mean square bound for the Mellin transform function of ζ1/2+ix4 is given.

Highlights

  • Power moments represent one of the most important parts of the theory of the Riemann zeta-function ζ(s), defined as ∞ ζ(s) = n−s n=1(σ = Re s > 1), and otherwise by analytic continuation

  • And a large literature exists on this subject

  • Ramachandra’s monograph [23]) one obtains that the expression in (1.3) is, for log log T ≪ G ≪ T, (1.5)

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Summary

Introduction

Power moments represent one of the most important parts of the theory of the Riemann zeta-function ζ(s), defined as. The Riemann zeta-function, power moments, asymptotic formulas, upper bounds. In the absence of asymptotic formulas for Ik(T ), one would like to obtain good upper bounds for Ik(T ). By its integral over a suitable (short) interval In employing this procedure one obviously loses something, but on the other hand one gains flexibility from the fact that explicit upper bound for Ik(T + G) − Ik(T − G) are known only in the case when k = 1 (see Lemma 1) and k = 2 (see [7, Theorem 5.2] and [22]). Ramachandra’s monograph [23]) one obtains that the expression in (1.3) is, for log log T ≪ G ≪ T , This shows that, up to ‘ε’, the bound in (1.4) is best possible. |Z2(1 + it)|2 dt ≪ε T 3/5+ε, while (2.14) improves both of these bounds, since 13/16 < 5/6 and 4/7 < 3/5

The necessary lemmas
The proof of Theorem 1
The proof of Theorem 3
Full Text
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