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

TL;DR

This paper advances bounds on Kloosterman sums with congruence conditions, enabling power-saving error terms in the dispersion method, with applications to divisor problems and the Titchmarsh divisor problem, under RH assumptions.

Contribution

It generalizes classical bounds on Kloosterman sums, leading to new power-saving error estimates in key number theory problems, including the Titchmarsh divisor problem and sums involving divisor functions.

Findings

Proved bounds for quintilinear sums of Kloosterman sums with congruence conditions.

Achieved power-saving error terms in the Titchmarsh divisor problem assuming RH.

Established power-saving in sums involving divisor functions, independent of k under generalized Lindelöf hypothesis.

Abstract

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. 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.