Friable averages of complex arithmetic functions (Q6619679)

Let \(P^+(n)\) denote the largest prime factor of a natural integer \(n>1\) with \(P^+(1)=1\), and \(S(x,y):=\{n\leqslant x:P^+(n)\leqslant y\}\) be the set of \(y\)-friable integers not exceeding \(x\). In the paper under review, the authors approximate friable sums\N\[\N\Psi(x,y;f):=\sum_{n\in S(x,y)}f(n)\N\]\Nfor complex arithmetical functions \(f\), whose Dirichlet series is analytically close to some complex power of the Riemann zeta function. The authors give some applications to the distribution of additive functions over \(S(x,y)\), more precisely to the small omega function \(\omega(n)=\sum_{p|n}1\), hence obtaining an Erdős-Kac type theorem for friable integers.