PROPRIÉTÉS LOCALES DES CHIFFRES DES NOMBRES PREMIERS. (4th April 2017)

• Record Type: Journal Article
• Authors: Martin, Bruno; Mauduit, Christian; Rivat, Joël
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 18:Number 1(2019)
• Page Start: 189
• Page End: 224
• Publication Date: 2017-04-04
• Subjects: 11A63, -- 11J71, -- 11K65, -- 11L20, -- 11N05; prime numbers, -- exponential sums, -- b-additive functions, -- uniform distribution modulo 1; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748017000044
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract : Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum $$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big), \end{eqnarray}$$ where the summation runs over prime numbers $p$ and where $\unicode[STIX]{x1D6FD}\in \mathbb{R}$, $k\in \mathbb{Z}$, and $g:\mathbb{N}\rightarrow \mathbb{Z}$ is a strongly $b$ -additive function such that $\operatorname{pgcd}(g(1), \ldots, g(b-1))=1$ .