Exponential sums with multiplicative coefficients and integers without large prime factor (Q1804720)

In Theorem 11 [Proc. Lond. Math. Soc., III. Ser. 63, 449-494 (1991; Zbl 0745.11042)], \textit{E. Fouvry} and \textit{G. Tenenbaum} obtained the limit as \(y\to \infty\) of the sum \({\displaystyle {\sum^\infty _{\substack{ n=1\\ P(n) \leq y}}}} n^{-1} e(n \theta)\), where \(P(n)= \max_{p|n} p\), \(P(1) =1\) and \(0<\theta< 1\). In this paper, the author considers a more general sum of the type \[ {1\over {\log y}} \sum^\infty_{\substack{ n=1\\ P(n) \leq y}} n^{-1} e(n\theta) f(n) \tag \(*\) \] as \(y\to \infty\), where \(f(n)\) is a complex-valued multiplicative function with \(|f(n) |\leq 1\) for all \(n\geq 1\). The main results show that the sum \((*)\) converges to 0 as \(y\to \infty\) for all irrational \(\theta\) and for those rational \(\theta\) for which certain series diverge, and an estimate for \((*)\) is derived for the remaining rational \(\theta\). The corresponding results for some special functions \(f\) are described, including \(f(n) =1\), \(\mu(n)\), \(\mu^2 (n)\),\(n^{-1} \varphi(n)\). For \(f(n)=1\) or \(\mu(n)\) and \(\theta= {a\over q}\) with \(q>1\), \((a,q)=1\), an estimate for \((*)\) is established that is uniform for \(q^{{1\over 2}+ \varepsilon} \log q\leq k\log y\), where \(\varepsilon >0\), \(k>0\).