On the simplicity defect of a natural number

Abstract

In the present paper we study elementary properties of the number-theoretic function x (n) introduced by us.The function x (n) (defined below) is related to the variational treatment of various problems of additive number theory. With the help of x (n) we construct inequalities which are satisfied for the whole set M of rational positive integers, but turn into equalities for some subsets of the set M. Thus the extreme attribution of prime numbers, prime pairs, etc. is carried out.