Study of the local law of \(\omega (n)\) in small intervals (Q1840493)

The author establishes an upper bound for the number of integers \(n\) with exactly \(k\) distinct prime factors and satisfying \(x<n\leq x+y\). His result is uniform for all positive \(k\) and \(x, y\) satisfying \(2\leq y\leq x\). When \(k<(1-\varepsilon)\;\frac{\log\log y}{\log \log \log y}\), his bound is of the same order of magnitude \[ \frac{y}{\log y} \frac{(\log \log y)^{k-1}}{(k-1)!} \] as the corresponding one for squarefree \(n\) obtained by \textit{R. Warlimont} and \textit{D. Wolke} [Math. Z. 155, 79-82 (1977; Zbl 0354.10037)]. The author proves his result using complicated but elementary combinatorial arguments, some of which depend on induction on \(k\). The constants he obtains at each stage can be given explicit values that lead to the values 6, 23 for the two constants appearing in the final result.