%0 Journal Article
%T Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method
%+ Institut de Mathématiques de Marseille (I2M)
%A Drappeau, Sary
%< avec comité de lecture
%@ 0024-6115
%J Proceedings of the London Mathematical Society
%I London Mathematical Society
%V 114
%N 4
%P 684-732
%8 2017
%D 2017
%Z 1504.05549
%R 10.1112/plms.12022
%Z Mathematics [math]/Number Theory [math.NT]Journal articles
%X 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.
%G English
%2 https://hal.science/hal-01302604/document
%2 https://hal.science/hal-01302604/file/1504.05549v3.pdf
%L hal-01302604
%U https://hal.science/hal-01302604
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-