Une inégalité de Hilbert pour les diviseurs. (A Hilbert inequality for divisors)
Let \(1=d_ 1<d_ 2<..<d_{\tau (n)}=n\) denote the sequence of divisors of n, and set \(H(n)=\sum_{1\leq i<j\leq \tau (n)}1/(d_ j-d_ i).\) An old conjecture of Erdős asserts that H(n) is of order O(\(\tau\) (n)). The author proves this conjecture in the stronger form \(H(n)\ll \tau (n)^{1-c}\), where c is an explicitly given positive constant. He further shows that the inequality \[ | \sum_{1\leq i<j\leq \tau (n)}\frac{\xi_ i\xi_ j}{d_ j-d_ i}| \leq \quad M\sum^{\tau (n)}_{i=1}\xi_ i^ 2\quad (\xi_ i\in {\mathbb{R}}), \] which is a special case of Hilbert's inequality, holds with a constant \(M=M(n)\) satisfying \(M(n)\ll \log \log \tau (n)\) as \(\tau\) (n)\(\to \infty\) and \(M(n_ i)\gg \log \log \tau (n_ i)\) for a suitable sequence \(\{n_ i\}\) with \(\tau (n_ i)\to \infty\) as \(i\to \infty\).
scientific article; zbMATH DE number 4202391