Problems similar to the additive divisor problem (Q1290820)

Under certain conditions to be imposed on the multiplicative function \(f\), the sum \[ S(f,x)=\sum_{n\leq x}f(n)\tau(n-1), \] where \(\tau\) is the divisor function, may be expected to have an asymptotic relation of the form \[ S(f,x)=C(f)\sum_{n\leq x}f(n)\log x(1+o(1)). \] The main result of the paper is that this is indeed the case when \(f\) is nonnegative and satisfies the following conditions: there exists \(A\) such that \(f(p^r)\leq A^r\) for primes \(p\) and \(r\geq 1\); for any \(\varepsilon>0\), there exists \(A_\varepsilon\) such that \(f(n)\leq A_\varepsilon n^\varepsilon\) for all \(n\); there exists \(\alpha>0\) such that, for large \(x\), \[ \sum_{p\leq x}f(p)\log p\geq\alpha x. \] The proof involves applications of the large sieve, and requires good uniform upper estimates for the average of \(f(n)\) in an arithmetic progression with a large modulus as given in a paper by \textit{P. Shiu} [J. Reine Angew. Math. 313, 161-170 (1980; Zbl 0412.10030)]. The explicitly determined valued for \(C(f)\) is too complicated to be reproduced here.