Hooley's \(\Delta _ r\)-functions when r is large (Q1074643)

In an important paper [Proc. Lond. Math. Soc., III. Ser. 38, 115-151 (1979; Zbl 0394.10027)], \textit{C. Hooley} introduced the functions \[ \Delta_ r(n)=\max_{u_ 1,...,u_{r-1}}card\{d_ 1,...,d_{r- 1}:\quad d_ 1...d_{r-1}| n,\quad u_ i<d_ i\leq eu_ i\quad (i=1,...,r-1)\}, \] and showed that estimates for \(\Delta_ r(n)\) have potential applications to various problems in additive number theory and diophantine approximation. Using Fourier techniques, Hooley established bounds of the type \((*)\quad \sum_{n\leq x}\Delta_ r(n) \ll x (\log x)^{\xi}\) with certain positive constants \(\xi =\xi (r).\) By estimating more carefully a multiple integral arising in Hooley's argument, the author improves on Hooley's values for the exponents in (*); for example, he shows that if \(4\leq r\leq 13\), then (*) holds with \(\xi =(r-1)^{3/2}/(r+\sqrt{r-1}).\) Remark: In the meantime, much stronger estimates have been obtained by the author and \textit{G. Tenenbaum} in a paper to appear in Compos. Math.).