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

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 ${\sum }_{p\le x}\tau (p-1)$. 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 ${\sum }_{n\le x}{\tau }_k(n)\tau (n+1)$, reproving a result announced by Bykovskiĭ and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindelöf hypothesis is assumed.

References in corpus (1)

• The Shifted Convolution of Divisor Functions

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• One-level density estimates for Dirichlet L-functions with extended support
• The fourth moment of individual Dirichlet L-functions on the critical line
• Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
• Chowla and Sarnak Conjectures for Kloosterman Sums
• Mean values of long Dirichlet polynomials with divisor coefficients
• The first moment of primes in arithmetic progressions: Beyond the Siegel-Walfisz range
• Kloosterman sums do not correlate with periodic functions
• Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$