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PROPRIÉTÉS LOCALES DES CHIFFRES DES NOMBRES PREMIERS

  • Bruno Martin (a1), Christian Mauduit (a2) and Joël Rivat (a3)

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 $\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$ .

Soit $b$ un nombre entier supérieur ou égal à 2. Nous donnons une formule asymptotique pour la somme d’exponentielles

$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$
où la sommation est effectuée sur les nombres premiers $p$ , et où $\unicode[STIX]{x1D6FD}$ est un nombre réel, $k$ un nombre entier et $g:\mathbb{N}\rightarrow \mathbb{Z}$ une fonction fortement $b$ -additive telle que $\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$ .

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Ce travail a bénéficié des aides de l’Agence nationale de la recherche portant les références « ANR-14-CE34-0009 » MUDERA, de Ciência sem Fronteiras, projet PVE 407308/2013-0, et du projet d’échange DynEurBraz (FP7 Irses 230844).

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References

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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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