A general sieve problem

Date/heure
5 mai 2022
14:30 - 15:30

Lieu
Salle Döblin

Oratrice ou orateur
Andreas Weingartner (Southern Utah University, États-Unis)

Catégorie d'évènement
Analyse et théorie des nombres

Given an arithmetic function $\theta$, we consider the set
$${\mathcal{B}}_{\theta }=\{n\ge 1:p|n\Rightarrow p\le \theta (\prod _{\genfrac{}{}{0ex}{}{q<p}{q^{\alpha }||n}}{q}^{\alpha })\},$$
where $p$ and $q$ denote primes. Depending on the choice of $\theta$, the possible sets ${\mathcal{B}}_{\theta }$ include the set of prime powers, almost primes, friable numbers, dense numbers, and practical numbers.
We will discuss (1) asymptotic results for the counting function of ${\mathcal{B}}_{\theta }$, (2) a generalization of the Siegel-Walfisz theorem, and (3) the normal order of the number of prime factors of integers in ${\mathcal{B}}_{\theta }$.