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H 1 On the local distribution of the product of two shifted primes and application

Auteurs : Deshouillers, Jean-Marc (Auteur de la Conférence)
CIRM (Editeur )

Résumé : T.C. Brown and A.R. Freedman proved that the set $\mathcal{P}_{2}$ of products of two primes contains no dense cluster; technically, $\mathcal{P}_{2}$ has a zero upper Banach density, defined as $\delta^{*}(\mathcal{P}_{2}) =\lim_{H\mapsto \infty} \limsup_{x\mapsto \infty} \frac{1}{H} Card \{n\in \mathcal{P}_{2}:x< n\leq x+H\}$.
Pramod Eyyunni, Sanoli Gun and I jointly studied the local behaviour of the product of two shifted primes $\mathcal{Q}_{2}=\{(q-1)(r-1):q,r \, primes\}$. Assuming a classical conjecture of Dickson, we proved that $\delta^{*}(\mathcal{Q}_{2}) = 1/6$. Notice that we know no un-conditional proof that $\delta^{*}(\mathcal{Q}_{2})$ is positive. The application, which was indeed our motivation, concerns the study of the local behaviour of the set $\mathcal{V}$ of values of Euler’s totient function. Assuming Dickson’s conjecture, we prove that $\delta^{*}(\mathcal{V})\geq 1/4$. The converse inequality $\delta^{*}(\mathcal{V})\leq 1/4$ had been proved in the previous millenium by K. Ford, S. Konyagin and C. Pomerance.

Keywords : Banach density; products of shifted primes; values of Euler’s function; Dickson’s conjecture

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 29/09/2020 Date de captation : 07/09/2020 Collection : Research talks ; Number Theory Durée : 00:27:25 Domaine : Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2020-09-07_Deshouillers.mp4 Informations sur la Rencontre Virtuelle Nom de la rencontre : Additive Combinatorics / Combinatoire additiveOrganisateurs de la rencontre : Balandraud, Eric ; Dousse, Jehanne ; Girard, Benjamin ; Schmid, Wolfgang ; Tringali, SalvatoreDates : 07/09/2020 - 11/09/2020 Année de la rencontre : 2020 URL Congrès : https://conferences.cirm-math.fr/2228.html Citation Data DOI : 10.24350/CIRM.V.19653603 Cite this video as: Deshouillers, Jean-Marc (2020). On the local distribution of the product of two shifted primes and application.CIRM . Audiovisual resource. doi:10.24350/CIRM.V.19653603 URI : http://dx.doi.org/10.24350/CIRM.V.19653603

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