A generalization of the Hardy–Ramanujan inequality and applications

Abstract

Let ω(n) denote the number of distinct prime factors of the positive integer n. In 1917, Hardy and Ramanujan showed that for all real numbers x≥2 and all positive integers k,∑n≤xω(n)=k1≤Cxlog⁡x(log⁡log⁡x+D)k−1(k−1)!, where C and D are absolute constants. We derive an analogous result when the summand 1 is replaced by f(n), for many nonnegative multiplicative functions f. Summing on k recovers a frequently-used mean-value theorem of Hall and Tenenbaum. We use the same idea to establish a variant of a theorem of Shirokov, concerning multiplicative functions that are o(1) on average at the primes.