English
The sum-of-digits function in linearly recurrent number systems and almost primes
Virtualconference
Auteurs : Madritsch, Manfred (Auteur de la conférence)
CIRM (Editeur )
11A63 - Radix representation; digital problems
11L07 - Estimates on exponential sums
11N05 - Distribution of primes
CIRM (Editeur )
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Résumé :
This is joint work with Jörg Thuswaldner from University of Leoben.
A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]
The $q$-adic number system corresponds to the linear recursion $G_{k+1}=qG_k$ and $G_0=1$. The first example of a real generalization is due to Zeckendorf who showed that the Fibonacci sequence $G_0=1$, $G_1=2$, $G_{k+2}=G_{k+1}+G_k$ for $k\geq0$ yields a representation for each positive integer. This is unique if we additionally suppose that no two consecutive ones exist in the representation. Similar restrictions hold for different recurrent sequences and they build the essence of these number systems.
In the present talk we investigate the representation of primes and almost primes in linear recurrent number systems. We start by showing the different results due to Fouvry, Mauduit and Rivat in the case of $q$-adic number systems. Then we shed some light on their main tools and techniques. The heart of our considerations is the following Bombieri-Vinogradov type result
\[\sum_{q < x^{\vartheta-\varepsilon}}\max_{y < x}\max_{1\leq a\leq q} \left\vert\sum_{\substack{n< y,s_G(n)\equiv b\bmod d\\ n\equiv b\bmod q}}1 -\frac1q\sum_{n < y,s_G(n)\equiv b\bmod d}1\right\vert \ll x(\log 2x)^{-A},\]
which we establish under the assumption that $a_1\geq30$. This lower bound is due to numerical estimations. With this tool in hand we are able to show that \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\]
Mots-Clés : sum of digits; linear recurrence number system; level of distribution
Codes MSC : A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]
The $q$-adic number system corresponds to the linear recursion $G_{k+1}=qG_k$ and $G_0=1$. The first example of a real generalization is due to Zeckendorf who showed that the Fibonacci sequence $G_0=1$, $G_1=2$, $G_{k+2}=G_{k+1}+G_k$ for $k\geq0$ yields a representation for each positive integer. This is unique if we additionally suppose that no two consecutive ones exist in the representation. Similar restrictions hold for different recurrent sequences and they build the essence of these number systems.
In the present talk we investigate the representation of primes and almost primes in linear recurrent number systems. We start by showing the different results due to Fouvry, Mauduit and Rivat in the case of $q$-adic number systems. Then we shed some light on their main tools and techniques. The heart of our considerations is the following Bombieri-Vinogradov type result
\[\sum_{q < x^{\vartheta-\varepsilon}}\max_{y < x}\max_{1\leq a\leq q} \left\vert\sum_{\substack{n< y,s_G(n)\equiv b\bmod d\\ n\equiv b\bmod q}}1 -\frac1q\sum_{n < y,s_G(n)\equiv b\bmod d}1\right\vert \ll x(\log 2x)^{-A},\]
which we establish under the assumption that $a_1\geq30$. This lower bound is due to numerical estimations. With this tool in hand we are able to show that \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\]
Mots-Clés : sum of digits; linear recurrence number system; level of distribution
11A63 - Radix representation; digital problems
11L07 - Estimates on exponential sums
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 : 27/11/2020 Sous Collection : Research talks Catégorie arXiv : Number Theory Domaine(s) : Théorie des Nombres Format : MP4 (.mp4) - HD Durée : 00:47:59 Audience : Chercheurs Download : https://videos.cirm-math.fr/2020-11-27_Madritsch.mp4 
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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 aléatoireOrganisateurs de la Rencontre : Rivat, Joël ; Tichy, Robert Dates : 23/11/2020 - 27/11/2020 Année de la rencontre : 2020 URL de la Rencontre : https://www.chairejeanmorlet.com/2256.html
Données de citation
DOI : 10.24350/CIRM.V.19687903Citer cette vidéo: Madritsch, Manfred (2020). The sum-of-digits function in linearly recurrent number systems and almost primes. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19687903 URI : http://dx.doi.org/10.24350/CIRM.V.19687903 |
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Bibliographie
- FOUVRY, E. et MAUDUIT, Christian. Méthodes de crible et fonctions sommes des chiffres. Acta Arithmetica, 1996, vol. 77, no 4, p. 339-351. - http://dx.doi.org/10.4064/aa-77-4-339-351
- FOUVRY, Etienne et MAUDUIT, Christian. Sommes des chiffres et nombres presque premiers. Mathematische Annalen, 1996, vol. 305, no 1, p. 571-599. - http://dx.doi.org/10.1007/BF01444238
- MAUDUIT, Christian et RIVAT, Joël. Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Annals of Mathematics, 2010, p. 1591-1646. - https://annals.math.princeton.edu/2010/171-3/10.4007/annals.2010.171.1591
