Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Sary Drappeau

Published 2015-04-21Version 1

Building on classical work of Deshouillers and Iwaniec and recent work of Blomer and Mili\'cevi\'c, we prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. We employ this estimate to obtain power-saving in the dispersion method, in the setting of Bombieri, Fouvry, Friedlander and Iwaniec. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\sum_{n\leq x}\tau_k(n)\tau(n+1)$, reproving a result announced by Bykovski\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\"of hypothesis is assumed.