On some mean value results for the zeta-function and a divisor problem II

Abstract

Let d(n) be the number of divisors of n, let Δ(x)≔∑n⩽xd(n)−x(logx+2γ−1) denote the error term in the classical Dirichlet divisor problem, and let ζ(s) denote the Riemann zeta-function. It is shown that ∫0TΔ(t)|ζ(12+it)|2dt≪T(logT)4. Further, if 2⩽k⩽8 is a fixed integer, then we prove the asymptotic formula ∫1TΔk(t)|ζ(12+it)|2dt=c1(k)T1+k4logT+c2(k)T1+k4+Oε(T1+k4−ηk+ε), where c1(k) and c2(k) are explicit constants, and where η2=3/20,η3=η4=1/10,η5=3/80,η6=35/4742,η7=17/6312,η8=8/9433. The results depend on the power moments of Δ(t) and E(T), the classical error term in the asymptotic formula for the mean square of |ζ(12+it)|.