ON THE DISTRIBUTION OF GENERATING FUNCTIONS

Abstract

Investigations concerning the generating function associated with the kth powers,

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originate with Hardy and Littlewood in their famous series of papers in the 1920s, ‘On some problems of “Partitio Numerorum”’ (see [7, Chapters 2 and 4]). Classical analyses of this and similar functions show that when P is large the function approaches P in size only for α in a subset of (0, 1) having small measure. Moreover, although it has never been proven, there is some expectation that for ‘most’ α, the generating function is about √P in magnitude. The main evidence in favour of this expectation comes from mean value estimates of the form

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An asymptotic formula of the shape (1.2), with strong error term, is immediate from Parseval's identity when s=2, and follows easily when s=4 and k>2 from the work of Hooley [2, 3, 4], Greaves [1], Skinner and Wooley [5] and Wooley [9]. On the other hand, (1.2) is false when s>2k (see [7, Exercise 2.4]), and when s=4 and k=2. However, it is believed that when t<k, the total number of solutions of the diophantine equation

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with 1[les ]x_j, y_j[les ]P (1[les ]j[les ]t), is dominated by the number of solutions in which the x_i are merely a permutation of the y_j, and the truth of such a belief would imply that (1.2) holds for even integers s with 0[les ]s<2k.

The purpose of this paper is to investigate the extent to which knowledge of the kind (1.2) for an initial segment of even integer exponents s can be used to establish information concerning the general distribution of f_P(α), and the behaviour of the moments in (1.2) for general real s.