A survey on \(k\)-freeness. (English) Zbl 1156.11338

Adhikari, S. D. (ed.) et al., Number theory. Proceedings of the international conference on analytic number theory with special emphasis on \(L\)-functions held at the Institute of Mathematical Sciences, Chennai, India, January 2002. Mysore: Ramanujan Mathematical Society (ISBN 81-902545-1-0/pbk). Ramanujan Mathematical Society Lecture Notes Series 1, 71-88 (2005).
Summary: We say that an integer \(n\) is \(k\)-free \((k\geq 2)\) if for every prime \(p\) the valuation \(v_p(n)<k\). If \(f:\mathbb N\to\mathbb Z\), we consider the enumerating function \({\mathcal S}_f^k(x)\) defined as the number of positive integers \(n\leq x\) such that \(f(n)\) is \(k\)-free. When \(f\) is the identity then \({\mathcal S}_f^k(x)\) counts the \(k\)-free positive integers up to \(x\). We review the history of \({\mathcal S}_f^k(x)\) in the special cases when \(f\) is the identity, the characteristic function of an arithmetic progression, a polynomial, arithmetic. In each section we present the proof of the simplest case of the problem in question using exclusively elementary or standard techniques.
For the entire collection see [Zbl 1057.11002].


11N25 Distribution of integers with specified multiplicative constraints