Moments of gaps between \(k\)-free numbers
Let \(k \geq 2\) be a fixed integer and let \(S_ n = S_ n(k)\) be the \(n\)-th \(k\)-free number. P. Erdős considered a problem of finding a largest \(\gamma\) such that \[ \sum_{s_ n\leq x}(s_{n+1} - s_ n)^ \gamma \ll x\tag{1} \] (obviously, if the inequality holds for \(\gamma_ 0\), it also holds for any \(\gamma \leq \gamma_ 0\)). Since the number of \(k\)-free integers \(\leq x\) is asymptotic to \(x/\zeta(k)\), (1) is the best possible. Erdős proved that if \(k=2\), then one can take \(\gamma = 2\). C. Hooley proved that for any \(k\) one can take \(\gamma = k+1\). The author of this paper improves Hooley's result by showing that (1) holds for any \(\gamma < 2k - 2 + 4/(k + 1)\).
scientific article; zbMATH DE number 205461