Title: On the monotonicity properties of additive representation functions
Abstract: If A is a set of positive integers, let R 1 ( n ) be the number of solutions of a + a ′ = n, a . a ′ ∈ A , and let R 2 ( n ) and R 3 ( n ) denote the number of solutions with the additional restrictions a < a ′ , and a ≤ a ′ respectively. The monotonicity properties of the three functions R 1 ( n ), R 2 ( n ), and R 3 ( n ) are studied and compared.