Pour $x$ assez grand, le nombre de diviseurs premiers de $n$ , $1 \leq n \leq x$ , a une distribution locale unimodale. Cela confirme une conjecture d’Erdös de 1948.

Comportement local moyen de la fonction de Brjuno

Michel Balazard; Bruno Martin — 2012

Fundamenta Mathematicae

We describe the average behaviour of the Brjuno function Φ in the neighbourhood of any given point of the unit interval. In particular, we show that the Lebesgue set of Φ is the set of Brjuno numbers and we find the asymptotic behaviour of the modulus of continuity of the integral of Φ.

On the variation of certain fractional part sequences

Michel Balazard; Leila Benferhat; Mihoub Bouderbala — 2021

Communications in Mathematics

Let $b > a > 0$ . We prove the following asymptotic formula $\sum_{n ⩾ 0} | {x / (n + a)} - {x / (n + b)} | = \frac{2}{π} ζ (3 / 2) \sqrt{c x} + O (c^{2 / 9} x^{4 / 9}),$ with $c = b - a$ , uniformly for $x ⩾ 40 c^{- 5} {(1 + b)}^{27 / 2}$ .

Étude d'une somme arithmétique multiple liée à la fonction de Möbius

Michel Balazard; Mongi Naimi; Y.-F. S. Pétermann — 2008

Acta Arithmetica

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