On the parity of additive representation functions

Let A be a set of positive integers, p(A, n) be the number of partitions of n with parts in A, and p(n) = p(N, n). It is proved that the number of n l ess than or equal to N for which p(n) is even is much greater than root N w hile the number of n less than or equal to N for which p(n) is odd is great er than or equal to N1/2+o(1) Moreover, by using the theory of modular form s, it is proved (by J.-P. Serre) that, for all a and m the number of n, suc h that n equivalent to a (mod m), and n less than or equal to N for which p (n) is even is greater than or equal to c root N for any constant c, and N large enough. Further a set A is constructed with the properties that p(A, n) is even for all n greater than or equal to 4 and its counting function A (x) (the number of elements of A not exceeding x) satisfies A(x) much great er than x/logx. Finally, we study the counting: Function of sets A such tha t the number of solutions of a + a' = n, a, a' epsilon A, a < a' is never 1 for large n. (C) 1998 academic Press.