CIRM - Videos & books Library - (Logarithmic) densities for automatic sequences along primes and squares

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Auteurs : Drmota, Michael (Auteur de la Conférence)
CIRM (Editeur )

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Résumé : It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has a logarithmic density -- and it can be decided when this logarithmic density is actually adensity. For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$. The purpose of this talk is to present a corresponding result for subsequences of general automatic sequences along primes and squares. This is a far reaching of two breakthroughresults of Mauduit and Rivat from 2009 and 2010, where they solved two conjectures by Gelfond on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ denotes thesequence of primes). More technically, one has to develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. Then asan application one can deduce that the logarithmic densities of any automatic sequence along squares $(n^2){n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, if densities exist then they are (usually) rational.

Keywords : automatic sequences; logarithmic densities; prime numbers

Codes MSC :
11A63 - Radix representation; digital problems
11B85 - Automata sequences
11L03 - Trigonometric and exponential sums, general
11L20 - Sums over primes
11N05 - Distribution of primes

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2020-11-23_Drmota.mp4

Informations sur la Rencontre

Nom de la rencontre : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire
Organisateurs de la rencontre : Rivat, Joël ; Tichy, Robert
Dates : 23/11/2020 - 27/11/2020
Année de la rencontre : 2020
URL Congrès : https://www.chairejeanmorlet.com/2256.html

Données de citation

DOI : 10.24350/CIRM.V.19686703
Citer cette vidéo: Drmota, Michael (2020). (Logarithmic) densities for automatic sequences along primes and squares. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19686703
URI : http://dx.doi.org/10.24350/CIRM.V.19686703

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Bibliographie

ADAMCZEWSKI, Boris, DRMOTA, Michael, et MÜLLNER, Clemens. (Logarithmic) densities for automatic sequences along primes and squares. arXiv preprint arXiv:2009.14773, 2020. - https://arxiv.org/abs/2009.14773

MAUDUIT, Christian, RIVAT, Joël, et al. La somme des chiffres des carrés. Acta Mathematica, 2009, vol. 203, no 1, p. 107-148. - http://dx.doi.org/10.1007/s11511-009-0040-0

MAUDUIT, Christian et RIVAT, Joël. Sur un probleme de Gelfond: la somme des chiffres des nombres premiers. Annals of Mathematics, 2010, p. 1591-1646. - https://www.jstor.org/stable/20752248

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