On the counts of p-rough numbers

Abstract

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line of symmetry for the function $\Phi(x,p)$, and we introduce the function $\Delta \Phi(x,p)$ which is the difference between $\Phi(x,p)$ and the line of symmetry. $\Delta \Phi(x,p)$ is periodic and bounded and has a rotational symmetry.