Note on the mean value of the Erd\H{o}s--Hooley Delta-function

For integer $n\geqslant 1$ and real $u$, let $\Delta(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erd\H{o}s--Hooley Delta-function is then defined by $\Delta(n):=\max_{u\in{\mathbb R}}\Delta(n,u).$ We improve a recent upper bound by Koukoulopoulos and Tao by showing that $\sum_{n\leqslant x}\Delta(n)\leqslant x(\log_2x)^{2+o(1)}$

Friable averages of oscillating multiplicative functions

We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some negative power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in the non-friable case. An application is given to summing truncated versions of such functions

Sommes de G\'al et applications

We evaluate the asymptotic size of various sums of G\'al type, in particular $$S( \mathcal{M}):=\sum_{m,n\in\mathcal{M}} \sqrt{(m,n) \over [m,n]},$$ where $\mathcal{M}$ is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for $$\log\Big( \sup_{|\mathcal{M}|= N}{S( \mathcal{M})/N}\Big)$$ and derive new lower bounds for localized extreme values of the Riemann zeta-function, for extremal values of some Dirichlet $L$-functions at $s=1/2$, and for large character sums.Comment: in French. v2: corrected quote (p.3) from Soundararajan (2008), due to a misprint in the published version of this article. v3: corrected an inaccuracy with no consequence on the statements; v4, v5: minor typos and inaccuracies corrected; v6: corrected an inaccuracy in the proof of thm 1.6; v7: final, accepted versio