A lower bound on the number of rough numbers

Abstract

Conceptually, a rough number is a positive integer with no small prime factors. Formally, for real numbers $x$ and $y$, let $\Phi (x,y)$ denote the number of positive integers at most $x$ with no prime factors less than $y$. In this paper we establish the lower bound $\Phi (n,p)\ge \lfloor 2n/p\rfloor +1$ when $p\ge 11$ is prime and $n\ge 2p$.

16 pages, appendix with 11 tables

References in corpus (1)

• Estimates of Some Functions Over Primes without R.H.