Results 1 
3 of
3
A SURVEY ON k–FREENESS
"... 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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
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. 1. Introduction The
Sieve Methods
"... called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their i ..."
Abstract
 Add to MetaCart
(Show Context)
called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their important applications in number theory. We apply sieves to study the distribution of squarefree numbers, smooth numbers, and prime numbers. The first chapter is a discussion of the basic sieve formulation of Legendre. We show that the distribution of squarefree numbers can be deduced using a squarefree sieve 1 . We give an account of improvements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun's Combinatorial sieve as presented in the modern language of [HR74]. We apply the general sieve to give a simpler
Abstract. Let k ∈ {3, 4, 5}. Let Rk(x) =
"... n≤x n is kfree 1 − x ζ(k) We give new upper bounds for Rk(x) conditional on the Riemann hypothesis, improving work of ..."
Abstract
 Add to MetaCart
(Show Context)
n≤x n is kfree 1 − x ζ(k) We give new upper bounds for Rk(x) conditional on the Riemann hypothesis, improving work of