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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 )

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    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

    Codes MSC :
    11B05 - Density, gaps, topology
    11B83 - Special sequences and polynomials
    11N32 - "Primes represented by polynomials; other multiplicative structure of polynomial values"
    11N64 - Other results on the distribution of values or the characterization of arithmetic functions


    Informations sur la Rencontre Virtuelle

    Nom de la rencontre : Additive Combinatorics / Combinatoire additive
    Organisateurs de la rencontre : Balandraud, Eric ; Dousse, Jehanne ; Girard, Benjamin ; Schmid, Wolfgang ; Tringali, Salvatore
    Dates : 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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