The Schnirelmann density of the squarefree integers
Kenneth Rogers · Proceedings of the American Mathematical Society · 1964
It is a familiar and elementary process to show that every natural number greater than one is the sum of two squarefree natural numbers: one shows that A (x)/x exceeds 1/2 for all x ? 1, where A (x) is the number of squarefree natural numbers not greater than x. This crude estimate follows from the fact that A (n) > n(1 p-2). It is also elementary that A(x)/x-*>6/wr2 as x>oo, and early numerical evidence might lead one to believe that 6/72 is also the Schnirelmann density of this sequence, the infimum of A(x)/x in the range 1<? x < m. The purpose of this note is to prove that this is not the case.