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
Résumé
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
$$
M_p(N) = \sum_{n \le N} \mu(n) \mathcal{K}_p(n) \qquad \mbox{and}\quad T_{\nu,p}(N) = \sum_{n \le N} \tau_\nu(n) \mathcal{K}_p(n)
$$
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
$$
M_p(N) \ll N \qquad \mbox{and}\quad T_{\nu,p}(N) \ll N (\log N)^{\nu -1}.
$$
for $N \ge p^{3/4+\varepsilon}$ and $N \ge p^{2/3+\varepsilon}$, respectively (for any fixed $\varepsilon>0$). We will explain the ideas of the recent joint work with Maxim Korolev (2020) where both these thresholds are lowered down to $N \ge p^{1/2+\varepsilon}$. We will also discuss some open questions.