On certain large additive functions

Abstract

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