On the Riemann zeta-function and the divisor problem

Abstract

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2 + it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) = - \Delta(x) +2\Delta(2x)- {1\over2}\Delta(4x)$, then we obtain $$ \int_0^T(E^*(t))^4 dt \ll_\epsilon T^{16/19+varepsilon}$$, 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 $|\zeta(1/2 + it)|$ over well-spaced points. This, in turn, yields a new proof of the twelfth moment estimate for $|\zeta(1/2 + it)|$. 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.