A uniform coprimality result for some arithmetic functions
Let \(f\) be an integer-valued multiplicative function such that for prime \(p\) we have \(f(p)=w(p)\) with a suitable polynomial \(w\), which is not of the form \(cx^ m\). Let \(v(p)\) be the number of non-zero solutions of \(w(x)\equiv 0 \pmod p\) and let \(d>1\) be a square-free integer, none prime divisor of which satisfies \(v(p)=p-1\) and which has a prime divisor \(q\) with \(v(q)\neq 0\). Under these conditions an asymptotic formula for \(N(x,d,f)\), the number of integers \(n\leq x\) satisfying \((d,f(n))=1\), is obtained with an error term, uniform for a certain range of \(d\). One of the results states that with a positive constant \(C(d)\) one has \[ N(x,d,f)=C(d)x(\log x)^{a(d)-1}(1+O_ M(\log^{-1/2} x)) \] for \(d\leq \log^ M x\) if there is a Siegel zero (mod \(d\)), resp. for \(d\leq \exp (A \log^{1/2} x)\) otherwise. Here \(A\) is a positive absolute constant. \(M\) is arbitrary and \(a(d)=\varphi^{-1}(d)\prod_{p \mid d}(p-v(p)-1)\).
scientific article; zbMATH DE number 3920589