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b-additive functions

Journal of the Institute of Mathematics of Jussieu (1)

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PROPRIÉTÉS LOCALES DES CHIFFRES DES NOMBRES PREMIERS
Part of
Bruno Martin, Christian Mauduit, Joël Rivat
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 1 / January 2019

Published online by Cambridge University Press:

04 April 2017, pp. 189-224

Print publication:

January 2019
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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 $\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$.