An all-purpose Erdös-Kac theorem

In a seminal paper of 1917, Hardy and Ramanujan showed that the normal number of prime factors of a random natural number n is log log n . Their paper is often seen as inspiring the development of probabilistic number theory in that it led Erdös and Kac to discover, in 1940, a Gaussian law implied by their work. In this paper, we derive an all-purpose Erdös-Kac theorem that is applicable in diverse settings. In particular, we apply our theorem to show the validity of an Erdös-Kac type theorem for the study of the number of prime factors of sums of Fourier coefficients of Hecke eigenforms.