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AUDIOVISUAL MATHEMATICS LIBRARY

AUDIOVISUAL MATHEMATICS LIBRARY

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The sum-of-digits function in linearly recurrent number systems and almost primes

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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 \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\]

Keywords : sum of digits; linear recurrence number system; level of distribution

Codes MSC :
11A63 - Radix representation; digital problems
 11L07 - Estimates on exponential sums
 11N05 - Distribution of primes
    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 01/12/2020
    Date de captation : 27/11/2020
    Sous collection : Research talks
    arXiv category : Number Theory
    Domaine : Number Theory
    Format : MP4 (.mp4) - HD
    Durée : 00:47:59
    Audience : Researchers
    Download : https://videos.cirm-math.fr/2020-11-27_Madritsch.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
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.19687903
Citer cette vidéo: (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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