The distribution in residue classes of integers omitting unspecified prime divisors (Q2910111)

During his career the author has considered various problems concerning the asymptotic behaviour of multiplicative functions under certain conditions; the reference section includes a long list of his relevant papers. In [J. Number Theory 12, 385--394 (1980; Zbl 0435.10028)] \textit{P. Erdős} and \textit{I. Z. Ruzsa} obtained a lower bound for the number of integers up to \(x\) divisible by no prime factor \(q\) in a set with \(\sum q^{-1}\leq K\). In this paper proofs are given of the author's previously unpublished results on integers lying in a congruence class and not divisible by any prime in some set with a known size but unspecified elements, with the aim of obtaining a result as uniform as possible. Let \(x\geq 3\), \(0<\varepsilon<{1\over 4}\) and denote by \(\mathop{{\sum}'}\) a summation over positive integers with no prime factor in a given set. Theorem 1 states that NEWLINE\[NEWLINE\mathop{{\sum}'}_{\substack{ n\leq x\\ n=a\pmod D}} 1={1\over \varphi(D)}\, \mathop{{\sum}'}_{\substack{ n\leq x\\ (n,D)= 1}} 1+ O\Biggl({x\over \varphi(D)} \Biggl({\log(D\log x)\over\log x}\Biggr)^{{1\over 4}-\varepsilon}\Biggr)NEWLINE\]NEWLINE uniformly for \((a,D)= 1\), \(D\geq 1\). Theorem 2 aims to obtain a formula for the sum on the right above with an \(O\)-term not involving \(D\). The proofs of both theorems utilize results from the literature. In both cases an error term involves the sum \(c= \sum q^{-1}\) over primes \(q\leq x\) belonging to the excluded set of primes. To eliminate \(c\), another version of these \(O\)-terms is found by applying Selberg's sieve, and then the relevant error terms are combined.