Smooth sums over smooth \(k\)-free numbers and statistical mechanics (Q2249145)

Let \(k\geqslant 2\) be an integer number. A number \(n\) is called \(k\)-free if \(p^k\nmid n\) for every prime number \(p\). The paper is devoted to the study of the asymptotic behavior of the following smooth sum \[ \sum_{\substack{ n \text{ is \(k\)-free} \\ p|n \;\Rightarrow\;p\leqslant N}} f\left(\frac{\log n}{\log N}\right)\frac{\alpha^{\Omega(n)}}{n} \] as \(N\rightarrow\infty\). Here \(k\geqslant 2\), \(\alpha\in \mathbb{C}\) and a bounded function \(f:\mathbb{R}\rightarrow\mathbb{C}\) are fixed, while \(\Omega(n)\) denote the number of prime divisors of \(n\), counted with multiplicity.