Distribution of squarefree integers in non-linear sequences

Ivan Ernest Stux

Vol. 59 (1975), No. 2, 577–584

DOI: 10.2140/pjm.1975.59.577

Abstract

I. This paper investigates the occurrences of the square-free integers in sequences s_n = [f(n)],n = 1,2,8,⋯ where f(x) belongs to classes of functions described by ‘smoothness’ conditions. The result obtained is an extention of the well known fact that Q(x) = 6∕π²x + O(X^1∕2), where Q(x) = number of squarefree integers ≦ x; it states that Q_s(x) ∼ 6∕π²g(x) where Q_𝜖(x) = number of squarefree integers ≦ x in the sequence s_n, and g(x) is the inverse function of f(x).

Mathematical Subject Classification

Primary: 10H20

Secondary: 10F40

Milestones

Received: 30 June 1975

Published: 1 August 1975