Let \(n\) be a positive integer, \(n=\prod\limits_{i=1}^rp_i^{\alpha_i}\) in canonical form, and let \(A(n)=\sum\limits_{i=1}^r\alpha_ip_i\). Clearly \(A\) is an additive arithmetic function. Assume the primes \(p_i\) are arranged so that \(p_1\leq p_2\sum\limits_{i=1}^r\alpha_i\). If \(f\) and \(g\) are arithmetic functions such that \(\sum\limits_{n\leq x}f(n)\sim\sum\limits_{n