Integers without divisors in a given progression
If follows from a result of Landau of 1909 that for any given integer \(q>2\), the number of positive integers \(n\leq x\) such that \(n\) has no prime factor congruent to \(1\pmod q\) is asymptotically equal to \(c(q)x/(\log x)^{1-1/\phi(q)}\) as \(x\) tends to infinity, with some well-understood positive constant \(c(q)\). Recently, Banks, Friedlander and the reviewer fixed an odd prime \(q\) and a residue class \(a\) coprime to \(q\) and considered the problem of estimating the number of \(n\leq x\) such that \(n\) has no divisor \(d>1\) in the congruence class \(a\pmod q\). They established that the number of such \(n\) is asymptotically equal to \(cx(\log\log x)^{\alpha}/(\log x)^{\beta}\) as \(x\) tends to infinity, with well-determined constants \(c>0,~\alpha\geq 0,~0<\beta\leq 1\) depending on \(a\) and \(q\). When \(a=1\), these constants are \(\alpha=q-3,~\beta=1\) and \(c=\frac{q\phi(q-1)}{(q-1)^{q-1)} (q-3)!}\). The purpose of the paper under review is two fold. Firstly, the authors show how one can simplify the somewhat ad-hoc analytical part of the argument due to Banks et. al. by using an approach used by Delange in several problems. Secondly, this more general approach allows the authors to deduce that similar estimates hold true when \(q\geq 3\) is an arbitrary modulus, not necessarily prime, but retaining the condition that \(a\) is coprime to \(q\). In this case, the values of the constants \(\alpha, ~\beta\) and \(c\) are more complicated since they depend on certain combinatorial constants associated to finite abelian groups. The approach is in fact even more general and also works to count integers \(n\leq x\) having no divisors \(d>1\) in a certain set of \(k\geq 2\) residue classes comprise to their modulus's, although in this later case the determination of \(\alpha,~\beta\) and \(c\) remains unachieved. It is likely that this nice paper will spur future research especially concerning the exact determination of the parameters \(\alpha,~\beta\) and \(c\) either in general or in interesting instances.
asymptotic results on arithmetic functions
scientific article; zbMATH DE number 5956828