The sum of \(\alpha^{\Omega(n)}\) over integers n\(\leq x\) with all prime factors between \(\alpha\) and y
Estimates of the function \(\psi_{\alpha}(x,y)=\sum \alpha^{\Omega(n)}\), \(\alpha\geq 1\), where the sum is taken over all n, 1\(\leq n\leq x\) such that if p is prime and \(p| n\) then \(\alpha<p\leq y\), are discussed. This function is related to the well known function \(\psi\) (x,y), which represents the number of \(n\leq x\) all of whose prime factors are \(\leq y\). The function by which \(\psi_{\alpha}(x,y)\) is approximated is \(x\rho_{\alpha}(u)\log^{\alpha -1}x\), where \(u=\log x/\log y\) and \(\rho_{\alpha}(u)\) is a solution of a certain differential delay equation. \(\{\) Reviewer's remark: in the definition of \(\rho_{\alpha}\) on p. 207 in ii) one should replace \(<\) by \(=\}\).
scientific article; zbMATH DE number 3841984