On certain large additive functions

Abstract

Let P(n) denote the largest prime factor of $n \ge 2, P(1) = 1$, and let $$ β(n) = \sum_{p|n}p, \Beta(n) = \sum_{p^α||n}αp, \Beta_1(n) = \sum_{\p^α||n}p^α$$ denote "large" additive functions. A survey of results on the functions $P(n), β(n), \Beta(n)$ and $\Beta_1(n)$ is presented, as well as some new results and open problems.