VALEURS IMPAIRES DE LA FONCTION DE PARTITION p(n)

Abstract

Let p(n) denote the number of partitions of n, and for i = 0 (resp. 1), Ai(x) denote the number of n ≤ x such that p(n) is even (resp. odd). In this paper, it is proved that for some constant K > 0, [Formula: see text] holds for x large enough. This estimation slightly improves a preceding result of S. Ahlgren who obtained the above lower bound for K = 0. Let [Formula: see text] and [Formula: see text]; the main tool is a result of J.-P. Serre about the distribution of odd values of τk(n). Effective lower bounds for A0(x) and A1(x) are also given.