On the Riemann zeta-function and the divisor problem

2004/04/30 by Aleksandar Ivić
Mathematics · #math.NT #msc:11N37 #msc:11M06

paper · pdf

published as Central European J. Math. 2(4) (2004), 1-15. · 18 pages

arxiv created 2004/07/02 · arxiv updated 2009/12/01

Abstract

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of |ζ(1/2 + it)|. If E^*(t) = E(t) - 2πΔ^*(t/(2π)) with Δ^*(x) = - Δ(x) +2Δ(2x)- 1\over2Δ(4x), then we obtain ∫₀^T(E^*(t))⁴ dt ≪_εT^{16/19+varepsilon}, which is the first non-trivial bound for higher moments of E^*(t). The method of proof also provides an upper bound for sums of fourth powers of mean square integrals of |ζ(1/2 + it)| over well-spaced points. This, in turn, yields a new proof of the twelfth moment estimate for |ζ(1/2 + it)|. Among the chief ingredients in the proof is a recent result of Robert--Sargos on the distribution of four square roots of integers, plus an approach of M. Jutila that involves the use of Airy integrals to deal with the ensuing exponential sums.

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