A simple proof that the square-free numbers have density 6/(pi^2)

Abstract

In this note we give a simple proof of the well-known result that the square-free numbers have density 6 ∕ π2.

Some observations on numbers in short intervals

“…There exist in the literature many works on the distribution of composite numbers in the interval [1, x] with their prime factors restricted in some form. Numbers with exactly k prime factors in their prime factorization (see, for example, [4]), square-free numbers and in general k-free numbers (see, for example, [7]), square-full numbers and in general k-full numbers (see, for example, [5]), smooth numbers (see, for example, [11]), etc. Also, there exist works on special prime factors.…”

Sums of fractional parts and sum of restricted divisors of a number

“…The sequence a of integers in arithmetic progression. The sequence a of h-free numbers (h ≥ 2), where A(x) ∼ 1 ζ(h) x (see, for example, [6]). In particular, for the sequence of square-free numbers we have A(x) ∼ 6 π 2 x.…”

Sequences related to the e number and Bernoulli numbers

“…The Euler's constant γ appear, for instance, in certain sums of fractional parts [4] [5]. The values of the zeta function ζ(s) also appear, for instance, in certain sums of fractional parts [3] and the counting function of many sequences of positive integers [2]. In this note, we study some sequences related simultaneously with the e number and the Bernoulli numbers.…”