On the divisor problem: Moments of Δ(x) over short intervals

Abstract

Let as usual ∆(x) denote the error term in the Dirichlet divisor problem, i.e., ∑ n≤x d(n) = x log x+ (2γ − 1)x+∆(x), x a large real variable and γ the Euler-Mascheroni constant. While asymptotics for the square-mean of ∆(x) are classic, K.-M. Tsang [Proc. London 1992] recently established results on the third and fourth power moments. We present short interval variants of such asymptotics. THEOREM 1. For T a large real variable, suppose that T 7→ Λ = Λ(T ) increases with T , satisfies 0 0, lim T→∞ T 1/2+ 0 Λ(T ) = 0. Then, as T →∞, ∫ T+Λ T−Λ ( ∆(t) )3 dt ∼ C3(λ)ΛT , and ∫ T+Λ T−Λ ( ∆(t) )4 dt ∼ C4(λ)ΛT , with explicite positive coefficients C3(λ), C4(λ).