Sums of nonnegative multiplicative functions over integers without large prime factors. II (Q2781454)

Let \(h\) denote a nonnegative multiplicative function such that there exist constants \(\delta\) \((0< \delta< 1)\), \(\kappa>1\), \(\varepsilon\) \((0< \varepsilon< \frac 12)\), \(b>0\) for which when \(z\geq 2\) NEWLINE\[NEWLINE\sum_{p\leq z}h(p)\log p= \kappa z+ O(z(\log z)^{-\delta}), \qquad \sum_{p, k\geq 2}h(p^k) p^{-k(1-\varepsilon)}\leq bNEWLINE\]NEWLINE hold (\(p\) being a prime). Let \(P(n)= \max \{p: p|n\}\), \(P(1)=1\), and \(S(x,y)= \{n\leq x: P(n)\leq y\}\). The objective of this interesting paper is to obtain an asymptotic formula for \(M(x,y)= \sum_{n\in S(x,z)} h(n)\) valid for \(u= \frac{\log x}{\log y}\) satisfying NEWLINE\[NEWLINE 1< u< (\log y)^{\delta/2}(\log\log y)^{-1}, \tag{*}NEWLINE\]NEWLINE for \(y\) sufficiently large. Similar sums were investigated by \textit{N. G. de Bruijn} and \textit{J. H. van Lint} in [Indag. Math. 26, 339-347, 348-359 (1964; Zbl 0131.28703)] under more general conditions than above. NEWLINENEWLINENEWLINEIn a previous paper [Acta Arith. 97, 329-351 (2001; Zbl 0985.11042)], the author studied the related sum \(m(x,y)= \sum_{n\in S(x,y)}h(n)n^{-1}\) under weaker conditions on \(h\) and with a longer range for \(u\). She utilizes the results of this earlier paper to prove by delicate arguments that, under the conditions stated above, NEWLINE\[NEWLINEM(x,y)= x(\log y)^{-1} V(y) \rho_\kappa(u) (1+O((\log y)^{-\delta/2} \log (u+1))), \tag{**}NEWLINE\]NEWLINE where \(V(y)= \prod_{p\leq y}(1+ \sum_{k=1}^\infty h(p^k)p^{-k})\) and \(\rho_\kappa(u)\) satisfies a certain differential-difference equation with delayed argument. Like the Dickman function \(\rho(u)\), \(\rho_\kappa(u)\) decreases very rapidly to 0 as \(u\to\infty\), and this influences the size of the error term. As a starting point of her proof, the author establishes a formula for \(M(x,y)\) with an error term that is weaker than that in (**) except for small \(u\) but which is valid for \(u\) in a range with a much larger upper bound than that in (*).