Generalization of a problem of Evelyn-Linfoot and Page in additive number theory

Abstract

Let k and l be integers such that 2⩽k ⩽l. Let Sk and Sl′lbe two subsets of positive integers with Sk⊒Qk and S′l ⊒ Ql, where Qk denotes the set of k-free integers. Further suppose that the characteristic functions of Sk and Sl′l are multiplicative. Let T(n) denote the number of representations of n in the form n=a+b, where a e Sk and b∈ Sl′l. In this paper we establish an asymptotic formula for T(n), when n is sufficiently large; and deduce several known asymptotic formulae as particular cases.