Global numerical bounds for the number-theoretic omega functions

Abstract

We obtain global explicit numerical bounds, with best possible constants, for the differences $\frac{1}{n}{\sum }_{k\le n}\omega (k)-\log \log n$ and$\frac{1}{n}{\sum }_{k\le n}\Omega (k)-\log \log n$, where $\omega (k)$ and $\Omega (k)$ refer to the number of distinct prime divisors, and the total number of prime divisors of $k$, respectively.