Multiplicative and Additive Problems of Partitions of Natural Numbers

Abstract

This article deals with the problem of multiplicative factorization of the natural number n on k factors and the additive partition of a natural number n on k terms provided that parameter k is a function of n and \(k \rightarrow \infty \) with \(n \rightarrow \infty \). We obtained asymptotic formulas for limiting cases of the order of growth of parameter k, which are characterized by the fact that the form of the asymptotic formula changes when k passes corresponding critical values \(k = k_{cr}(n)\). This feature occurs in additive and multiplicative problems of the partition (factorization) of natural numbers. As an application, it is noted the point of maximum of the function of additive partition into unordered terms of interest to the critical state in Maslov’s model of Bose-condensate, which built a new distribution corresponding to the real noble gas and the equation of state for him. Another application is the new fast algorithms for computing multiplicative and additive functions of partitions with different conditions on parameters n and k.