On Ingham's Tauberian theorem for partitions

Abstract

The object of this paper is to extend the range of validity of Ingham's Tauberian theorem for partitions (2). Ingham's theorem deals with many types of partitions, but it cannot be used in some cases of importance. With a slight modification of Ingham's notation we definewhere P(u) is the number of solutions ofin integers ni ≤ 0, and the λi are a given set of numbers, not necessarily integers, such that 0 < λ1 < λ2 <.… If N(u), the number of λi not exceeding u, is given bywhere B > 0, β > 0, and as u → ∞then, according to Ingham's theorem, when u → ∞where