arXiv 1909.084992019-09-21math.NT

线性递归数列的位数和函数的水平分布

The level of distribution of the sum-of-digits function of linear recurrence number systems

Manfred G. Madritsch, Jörg M. Thuswaldner

备注：: 22 pages, 1 figure, 1 table

摘要：

设$G=(G_j)_{j\ge 0}$是严格递增线性递归整数序列，且$G_0=1$具有特征多项式$X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$ .众所周知，对于满足$G_\ell \le ν< G_{\ell+1}$的$\ell \in \mathbb{N}$，每个正整数$ν$可以由所谓的贪婪展开$ν=\varepsilon_0(ν)G_0+\cdots+\varepsilon_\ell(ν)G_\ell$唯一地表示。这里的数字是以$0\le ν- \varepsilon_{\ell}(ν) G_\ell - \cdots - \varepsilon_j(ν) G_j < G_j$适用于$0 \le j \le \ell$的方式递归定义的。本文研究了序列$G$上在某些自然假设下的位数和函数$s_G(ν)=\varepsilon_0(ν)+\cdots+\varepsilon_\ell(ν)$。特别是，我们确定其分布水平$x^{\vartheta}$。为了更精确，我们表明，对于$r,s\in\mathbb{N}$与$\gcd(a_1+\cdots+a_d-1,s)=1$，我们有每个$x\ge 1$和所有$A,\varepsilon\in\mathbb{R}_{>0}$，\[ \sum_{q<x^{\vartheta-\varepsilon}}\max_{z<x}\max_{1\leq h\leq q} \lvert\sum_{\substack{k<z,s_G(k)\equiv r\bmod s\\ k\equiv h\bmod q}}1 -\frac1q\sum_{k<z,s_G(k)\equiv r\bmod s}1\rvert \ll x(\log 2x)^{-A}. \]这里$\vartheta=\vartheta(G) \ge \frac12$可以显式计算，我们有$\vartheta(G) \to 1$的$a_1\to\infty$。作为一个应用，我们证明了$\#\{ k\le x \,:\, s_G(k) \equiv r \pmod{s}, \; k \hbox{ has at most two prime factors} \} \gg x/\log x $提供的系数$a_1$是不是太小。此外，使用Besieri的筛的“几乎素数定理”$s_G$如下从我们的结果。我们的工作扩展了早期的经典$q$ -ary和的数字函数Fouvry和Mauduit获得的结果。

英文摘要：

Let $G=(G_j)_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each positive integer $ν$ can be uniquely represented by the so-called greedy expansion $ν=\varepsilon_0(ν)G_0+\cdots+\varepsilon_\ell(ν)G_\ell$ for $\ell \in \mathbb{N}$ satisfying $G_\ell \le ν< G_{\ell+1}$. Here the digits are defined recursively in a way that $0\le ν- \varepsilon_{\ell}(ν) G_\ell - \cdots - \varepsilon_j(ν) G_j < G_j$ holds for $0 \le j \le \ell$. In the present paper we study the sum-of-digits function $s_G(ν)=\varepsilon_0(ν)+\cdots+\varepsilon_\ell(ν)$ under certain natural assumptions on the sequence $G$. In particular, we determine its level of distribution $x^{\vartheta}$. To be more precise, we show that for $r,s\in\mathbb{N}$ with $\gcd(a_1+\cdots+a_d-1,s)=1$ we have for each $x\ge 1$ and all $A,\varepsilon\in\mathbb{R}_{>0}$ that \[ \sum_{q<x^{\vartheta-\varepsilon}}\max_{z<x}\max_{1\leq h\leq q} \lvert\sum_{\substack{k<z,s_G(k)\equiv r\bmod s\\ k\equiv h\bmod q}}1 -\frac1q\sum_{k<z,s_G(k)\equiv r\bmod s}1\rvert \ll x(\log 2x)^{-A}. \] Here $\vartheta=\vartheta(G) \ge \frac12$ can be computed explicitly and we have $\vartheta(G) \to 1$ for $a_1\to\infty$. As an application we show that $\#\{ k\le x \,:\, s_G(k) \equiv r \pmod{s}, \; k \hbox{ has at most two prime factors} \} \gg x/\log x $ provided that the coefficient $a_1$ is not too small. Moreover, using Bombieri's sieve an "almost prime number theorem" for $s_G$ follows from our result. Our work extends earlier results on the classical $q$-ary sum-of-digits function obtained by Fouvry and Mauduit.

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