Sums of nonnegative multiplicative functions over integers without large prime factors. I (Q2717605)

Let \(h\) denote a nonnegative multiplicative function satisfying specific conditions that imply that on average \(h\) is about \(\kappa\) on primes and that its magnitude on prime powers is restricted. In this nicely written and interesting paper the author derives an asymptotic formula with an error term for the sum \(\sum_{n\in S(x,y)} n^{-1}h(n)\) where \(S(x,y)= \{n\in \mathbb{N}: n\leq x,P(n)\leq y\}\) with \(P(n)= \max_{p|n}p\). Her result extends and improves one obtained by \textit{N. G. de Bruijn} and \textit{J. H. van Lint} [Indag. Math. 26, 339-347, 348-359 (1964; Zbl 0131.28703)]. She states, proves and uses some unpublished work of H. Halberstam on this sum. Her proofs are intricate and technical although elementary. An alternative but overlapping approach is given by \textit{G. Tenenbaum} in [Acta Arith. 97, 353-360 (2001; Zbl 0985.11043)].