Investigations concerning the generating function associated with
the kth powers,
formula here
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
formula here
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
formula here
with 1[les ]xj,
yj[les ]P (1[les ]j[les ]t),
is dominated by the number of solutions in which the
xi are merely a permutation of the
yj, 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
fP(α), and the behaviour of the
moments in (1.2) for general real s.