×

Entiers sans grand facteur premier en progressions arithmetiques. (Integers free of large primefactors in arithmetic progressions). (French) Zbl 0745.11042

Let \(S(x,y)\) be the set of positive integers \(\leq x\) that are free of prime factors \(>y\). Denote as usual by \(\Psi(x,y)\) the cardinality of \(S(x,y)\), and, for positive integers \(a\) and \(q\), define \(\Psi_ q(x,y)\) and \(\Psi(x,y;a,q)\), respectively, as the number of integers in \(S(x,y)\) that are coprime to \(q\), and the number of integers in \(S(x,y)\) that are congruent to \(a\) modulo \(q\). In the paper under review, the authors establish several estimates for these functions that in part represent analogs of well-known results on the distribution of primes in arithmetic progressions. In particular, the authors prove a Siegel-Walfisz type theorem for \(\Psi(x,y;a,q)\) for the range \(q\leq(\log x)^ A\), \((a,q)=1\), and a Bombieri-Vinogradov type estimate of the form \[ \sum_{q\leq\sqrt{x}(\log x)^{-B}}\max_{z\leq x}\max_{(a,q)=1} \left|\Psi(z,y;a,q)-{{\Psi_ q(z,y)}\over{\phi(q)}}\right|\;\ll\;{x\over (\log x)^ A}, \] which is valid uniformly in \(x\geq y\geq 2\), for any fixed constant \(A\) and a suitable constant \(B=B(A)\). They apply these results to obtain an estimate for exponential sums over integers in \(S(x,y)\) which is analogous to Vinogradov’s bound for exponential sums over primes.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11N35 Sieves
11L07 Estimates on exponential sums
PDFBibTeX XMLCite
Full Text: DOI