Entiers ultrafriables en progressions arithmétiques

Abstract

A natural integer is called $y$-ultrafriable if none of the prime powers occurring in its canonical decomposition exceed $y$. We investigate the distribution of $y$-ultrafriable integers not exceeding $x$ among arithmetic progressions to the modulus $q$. Given a sufficiently small, positive constant $\epsilon$, we obtain uniform estimates valid for $q⩽y^{c/{\log }_{2}y}$ whenever $\log y⩽(\log x{)}^{\epsilon }$, and for $q⩽\sqrt{y}$ if $(\log x{)}^{2+\epsilon }⩽y⩽x$.

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