An inequality related to the sieve of Eratosthenes

Abstract

Let $\Phi (x,y)$ denote the number of integers $n\in [1,x]$ free of prime factors $\le y$. We show that but for a few small cases, $\Phi (x,y)<.6x/\log y$ when $y\le \sqrt{x}$.

arXiv comment15 pages, 1 figure; to appear in the Journal of Number Theory