On a Result of Hardy and Ramanujan

TL;DR

This paper presents explicit approximations for the sum of the total number of prime factors of integers up to n, building on classical results by Hardy and Ramanujan.

Contribution

It introduces new explicit formulas for summing the total prime factors function, extending previous theoretical work by Hardy and Ramanujan.

Findings

Derived explicit approximation formulas for a ext{sum}_{k \u2264 n} \u03a9(k)

Improved understanding of the distribution of prime factors

Enhanced computational methods for prime factorization sums

Abstract

In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}\Omega(k)$, where $\Omega(k)$ is the total number of prime factors of $k$.