Mean values of multiplicative functions on the set
of \({\mathcal P}_k+1\) where \({\mathcal P}_k\) runs over the integers
having \(k\) distinct prime factors

Abstract. We investigate the limit behaviour of \[ \sum_{\substack{n\leq x\\[1ex] n\in \mathcal P_k}}g(n+1) \] as \(x\) tends to infinity where \(g\) is multiplicative with values in the unit disc and \(\mathcal P_k\) runs over the integers having \(k\) distinct prime factors. We let \(k\) vary in the range \(2\leq k\leq \epsilon(x)\log\log x\) where \(\epsilon(x)\) is an arbitrary function tending to zero as \(x\) tends to infinity.