Correlations between Kloosterman sums and multiplicative functions.

3:00pm, Wednesday, 11th October

Abstract

We consider Kloosterman sums $K(n) = \sum_{x=1}^{p-1} \exp(2 \pi i (nx + x^{-1})/p)$ modulo a prime $p$ and define their correlations

$M(N) = \sum_{n \le N} \mu(n) K(n)$ and $T(N) = \sum_{n \le N} \tau(n)  K(n)$,

with the Mobius  and  divisor functions.

Fouvry, Kowalski & Michel (2014) and Kowalski, Michel & Sawin (2018) improved the trivial bounds

$M(N) \ll N$  and  $T(N) \ll N (\log N)$

for $N  \sim  p^{3/4}$ and $N \sim p^{2/3}$, respectively. We will explain the ideas of recent joint work with Maxim Korolev where both these thresholds are both lowered down to $N \sim p^{1/2}$. 

We will also discuss some open questions.