Sums of Kloosterman sums with multiplicative coefficients

Date/heure
23 juin 2022
14:00 - 15:00

Lieu
Salle Döblin

Oratrice ou orateur
Igor Shparlinski (University of New South Wales)

Catégorie d'évènement
Analyse et théorie des nombres

We consider Kloosterman sums
$$K_p(n)=\sum _{x=1}^{p-1}\exp (2\pi i(nx+x^{-1})/p)$$
modulo a prime $p$ and define their sums
$$M_p(N)=\sum _{n\le N}\mu (n){\mathcal{K}}_p(n)\text{and}{T}_{\nu ,p}(N)=\sum _{n\le N}{\tau }_{\nu }(n){\mathcal{K}}_p(n)$$
twisted by the Möbius function $\mu (n)$ and by the $\nu$-fold divisor function ${\tau }_{\nu }(n)$. Fouvry, Kowalski & Michel (2014) and Kowalski, Michel & Sawin (2018) improved the trivial bounds
$$M_p(N)\ll N\text{and}{T}_{\nu ,p}(N)\ll N(\log N{)}^{\nu -1}.$$
for $N\ge p^{3/4+\epsilon }$ and $N\ge p^{2/3+\epsilon }$, respectively (for any fixed $\epsilon >0$). We will explain the ideas of the recent joint work with Maxim Korolev (2020) where both these thresholds are lowered down to $N\ge p^{1/2+\epsilon }$. We will also discuss some open questions.