Virtualconference
H (Logarithmic) densities for automatic sequences along primes and squares
Auteurs : Drmota, Michael (Auteur de la Conférence)
CIRM (Editeur )
CIRM (Editeur )
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- 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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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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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 01/12/2020 Date de captation : 23/11/2020 Collection : Research talks ; Computer Science ; Number Theory Durée : 00:46:04 Domaine : Computer Science ; Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2020-11-23_Drmota.mp4 
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Informations sur la Rencontre Virtuelle
Nom de la rencontre : Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoireOrganisateurs 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
Citation Data
DOI : 10.24350/CIRM.V.19686703Cite this video as: 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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