Sur la concentration moyenne des diviseurs. (On the mean concentration of divisors)
Let \[ \Delta (n)=\max_{u}card\{ d| n:\quad u<\log d\leq u+1\},\quad S(x)=\sum_{n\leq x}\Delta (n). \] In a well-known paper [Proc. Lond. Math. Soc., III. Ser. 38, 115-151 (1979; Zbl 0394.10027)], \textit{C. Hooley} proved that \((*)\quad S(x)\ll x(\log x)^{\lambda}\) holds with \(\lambda =(4/\pi)-1=0.273...\) and applied this estimate to problems in additive number theory and diophantine approximation. R. R. Hall and the author reduced the exponent in (*) to \(\lambda =0.238..\). and more recently to \(\lambda =0.219... \). In the paper under review, the author establishes (*) for every \(\lambda\) ; more precisely, he shows that, with a suitable constant c, \[ S(x)\ll x \exp \{c\sqrt{(\log \log x)(\log \log \log x)}\}. \] The proof is elementary, though not simple. The idea is to approximate \(\Delta\) (n) by high-order moments of the function \(\Delta (n,u)=card\{d| n:\quad u<\log d\leq u+1\}\) and evaluate \(\Delta\) (n,u) recursively by means of the identity \[ \Delta (n,u)=\Delta (n/p,u)+\Delta (n/p,u-\log p)\quad (p \| n). \]
scientific article; zbMATH DE number 3929096