Optimality for Tauberian theorems
Catégorie d'évènement : Analyse et théorie des nombres Date/heure : 22 juin 2022 10:00-11:00 Lieu : Salle Döblin Oratrice ou orateur : Gregory Debruyne (Ghent University) 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.
Well-behaved Beurling number systems
Catégorie d'évènement : Analyse et théorie des nombres Date/heure : 22 juin 2022 11:00-12:00 Lieu : Salle Döblin Oratrice ou orateur : Frederik Broucke (Ghent University) Résumé :A Beurling number system generalizes the multiplicative structure of the classical primes and integers. It consists of a non-decreasing unbounded sequence of real numbers $\{p_j\}_{j=1}^{\infty}$ with $p_1>1$, called the generalized primes, and the sequence of generalized integers $\{n_k\}_{k=0}^{\infty}$ which consists of the number 1 and all possible products of (powers of) the $p_j$. With such a system, one associates counting functions $\pi(x)$ and $N(x)$, counting the number of generalized primes and integers, respectively, below $x$. The primes satisfy the PNT if $\pi(x) \sim x/\log x$, and the integers have a density if $N(x) \sim \rho x$ for some positive $\rho$. If in these relations one has an error term of the form $O(x^a)$ for some $a<1$, one calls the primes or integers well-behaved.
In this talk, I will discuss various properties of these classes of Beurling systems, including extremal examples and omega results. I also discuss systems for which the primes and integers are simultaneously well-behaved. Finally, I will talk about some open problems.