Moderate and Large Deviations for the Erdős–kac Theorem

Abstract

Abstract. Erdős-Kac theorem is a celebrated result in number theory which says that the number of distinct prime factors of a uniformly chosen random integer satisfies a central limit theorem. In this paper, we establish the large deviations and moderate deviations for this problem in a very general setting for a wide class of additive functions.

Large and moderate deviation principles for alternating Engel expansions

“…For example, Mehrdad and Zhu [9] established the large and moderate deviations for Erdős-Kac theorem, which is a celebrated result about the number of distinct prime factors of a uniformly chosen random integer in number theory. Later, Zhu [18] and Hu [5] studied, respectively, the large deviations and moderate deviations for Engel, Sylvester and Cantor product expansions considered by Erdős et al [2], which are the classical representations of real numbers in number theory.…”

The Eigenvalue Point Process for Symmetric Group Permutation Representations on $k$-tuples

“…For m ≥ 2 and any δ > 0, Note that this is a bound for each m rather than an asymptotic bound. For more information on Erdős-Kac large deviation results, see for instance [17,24,14,23].…”

Large and Moderate Deviation Principles for Engel Continued Fractions

“…The authors in [13] considered two interesting discrete Markov processes introduced by Williams [41], which share the same classical limit theorems but have a difference in the context of large deviations. Besides, the theory of large deviations also has been applied to the analytic number theory, see Féray et al [15], Giuliano and Macci [18], Mehrdad and Zhu [28,29] and Radziwill [32]. It seems that the large and moderate deviations might have the potential to become the useful tools in studying probabilistic and analytic number theory.…”