Sandro Bettin, Università degli studi di Genova (Italie)


High moments of the Estermann function

For a rational number h/k, the Estermann fucntion is defined as the Dirichlet series D(s, h/k) = Σn ≥ 1 d(n) einh/k / ns for R(s) > 1 and by meromorphic continuation in the rest of the complex plane. We will show how to compute all moments of the Estermann function at the central point s = 1/2 when averaging over h modulo k as k goes to infinity among primes. In doing so we are also led to the computation of the asymptotic with power saving error term for the number of points in the projective variety x0 y0 + ... + xm ym = 0.