Moderate and Large Deviations for Erdős-kac Theorem

as n → ∞. This is a deep extension of Hardy-Ramanujan Theorem (see [13]). In addition, the central limit theorem holds in a more general setting for additive functions. A formal treatment and proofs can be found in e.g. Durrett [7]. In terms of rate of convergence to the Gaussian distribution, i.e. Berry-Esseen bounds, Rényi and Turán [18] obtained the sharp rate of convergenceO(1/ √ log logn). Their proof is based on the analytic theory of Dirichlet series. Recently, Harper [14] used a more probabilistic approach and used Stein’s method to get an upper bound of rate of convergence of the order O(log log logn/ √ log logn). In terms of large deviations, Radziwill [17] used analytic number theory approach to get a series of asymptotic estimates. Féray et al. [12] proved precise large deviations using the mod-Poisson convergence method developed in Kowalski and Nikeghbali [16]. In this paper, we study the large deviation principle and moderate deviation principle in the sense of Erdős and Kac. Instead of precise deviations that have been studied in Féray et al. [12]. Our large deviations and moderate deviations results are in the sense of Donsker-Varadhan [19], [3], [4], [5], [6]. Our proofs are probabilistic and require only elementary number theory results, in constrast to the analytical number theory approach in Radziwill [17]. Both large deviations and moderate deviations in our paper are proved for a much wider class of additive functions than what have been studied in Radziwill [17] and Féray et al. [12].