Frequency of coprimality of the values of a polynomial and a prime-independent multiplicative function (Q1081634)

Let \(P\) be a polynomial with integer coefficients, \(H\), \(J\) be positive integers, and \(M\) be a nonzero integer-valued multiplicative function such that, for all primes \(p\), \(M(p)=H\), \(M(p^2)=J\). The author proves that \[ \#\{n\leq x: (P(n),M(n))=1\} = Cx + O(x^{1/2}(\log x)^{2J} E(x,M)) \tag{1} \] where the constant \(C\) depends on \(M\) and \(P\), and \(E(x,M)\) is a certain sum over cubefull integers. In particular, she considers some interesting special cases: (i) \(P(n)=n\); (ii) \(M(n)=d_k(n)\), the number of ways of writing \(n\) as a product of \(k\) positive integer factors (so that \(d_2(n)\) is the usual divisor function \(d(n))\), in which case the sum \(E(x,d_k)\) in (1) can be omitted. In case (i), the main term in (1), without an estimate of the error term, was obtained by the reviewer [Acta Arith. 29, 401--423 (1976; Zbl 0286.10023)].