We say that an integer n is k–free (k 2) if for every prime p the valuation vp(n) < k. If f : N ! Z, we consider the enumerating function Sk f (x) defined as the number of positive integers n x such that f(n) is k–free. When f is the identity then Sk f (x) counts the k–free positive integers up to x. We review the history of Sk f (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.
Pappalardi, F. (2005). A SURVEY ON $K$-FREENESS. In R.B. S.D. Adhikari (a cura di), NUMBER THEORY - RAMANUJAN MATH. SOC. LECT. NOTES SER (pp. 71-88). MYSORE : RAMANUJAN MATH. SOC..
A SURVEY ON $K$-FREENESS
PAPPALARDI, FRANCESCO
2005-01-01
Abstract
We say that an integer n is k–free (k 2) if for every prime p the valuation vp(n) < k. If f : N ! Z, we consider the enumerating function Sk f (x) defined as the number of positive integers n x such that f(n) is k–free. When f is the identity then Sk f (x) counts the k–free positive integers up to x. We review the history of Sk f (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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
