# ANNALES DE L'INSTITUT FOURIER

On the fractional parts of $x/n$ and related sequences. II
Annales de l'Institut Fourier, Volume 27 (1977) no. 2, pp. 1-30.

As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of $xh\left(x\right)$ where $h$ is an arithmetical function (namely $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) and $n$ is an integer (or a prime order) running over the interval $\left[y\left(x\right),x\right)\right]$. The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

Comme promis dans l’article no I de même titre (Ann. Inst. Fourier, 26-4 (1976), 115-131), nous étudions ici la répartition asymptotique des parties fractionnaires de $xh\left(n\right)$$h$ est une fonction arithmétique (à savoir $h\left(n\right)=1/n$, $h\left(n\right)=logn$, $h\left(n\right)=1/logn$) et $n$ un entier (ou un nombre premier) parcourant l’intervalle $\left[y\left(x\right),x\right)\right]$. On s’est efforcé de démontrer des formes assez fines des théorèmes, encore que certains résultats se prêtent à des améliorations au prix d’une technicité accrue. Des applications arithmétiques seront données plus tard.

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Saffari, Bahman; Vaughan, R. C. On the fractional parts of $x/n$ and related sequences. II. Annales de l'Institut Fourier, Volume 27 (1977) no. 2, pp. 1-30. doi : 10.5802/aif.649. https://aif.centre-mersenne.org/articles/10.5802/aif.649/

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