Uniform estimates for certain multiplicative properties (Q790160)

Let \(a(n)\) denote a multiplicative arithmetic function taking the values \(0,1\) only, and suppose that the set \(\mathcal P\) of primes \(p\) with \(a(p)=1\) has density \(\alpha(\mathcal P)\) \((0<\alpha(\mathcal P)<1)\). Using sieve results, upper and lower bounds, for the same order of magnitude apart from a factor \(\alpha(\mathcal P)\), are obtained for \(\#\{n\leq x : a(n)=1\}\) in terms of \(x\) and \(\mathcal P\), these bounds being otherwise independent of the function \(a(n)\). If \(f\) is a multiplicative function in a class containing Euler's function and the divisor functions, the general results are applied to obtain close bounds for \[ N(x;d,f)=\#\{n\leq x : (d,f(n))=1\} \] that hold uniformly for squarefree \(d\) satisfying a further condition and such that the largest prime factor of \(d\) does not exceed \(x^{\delta}\) for a fixed \(\delta\) with \(0<\delta<1\). These bounds throw more light on the size (in terms of \(d\) and \(x\)) of the main term of the asymptotic formula for \(N(x;d,f)\) established by analytic methods in an earlier paper by the same author [J. Number Theory 20, 315--353 (1985; Zbl 0576.10030)] and valid uniformly for \(d\) in a smaller range than that given above.