Date/heure
22 juin 2022
10:00 - 11:00
Lieu
Salle Döblin
Oratrice ou orateur
Gregory Debruyne (Ghent University)
Catégorie d'évènement Analyse et théorie des nombres
Résumé
One version of the Ingham-Karamata theorem states that for each slowly oscillating function $\tau$ whose Laplace transform admits an analytic continuation beyond the line $\Re s \: s = 0$ must obey the asymptotic law $\tau(x) = o(1)$. This theorem is a cornerstone in Tauberian theory and has plenty of applications in number theory; one of the quickest proofs of the Prime Number Theorem passes through this theorem.
We shall show that the decay rate $o(1)$ in the Ingham-Karamata theorem is optimal even if one assumes analytic continuation of the Laplace transform up to a larger halfplane. The attractive proof is based on the open mapping theorem.