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V
The Waring Problem

In a famous letter to Euler, Waring wrote his great conjecture about
sums of powers. Lagrange had already proved his magnificent the-
orem that every positive integer was the sum of four squares, and
Waring guessed that this was not just a property of squares, but that,
in fact, the sum of a fixed number of cubes, fourth powers, fifth pow-
ers, etc., also worked. He guessed that every positive integer was
the sum of 9 cubes, 19 fourth powers, 37 fifth powers, and so forth,
and although no serious guess was made as to how the sequence 4
(squares), 9, 19, 37, ... went on, he simply stated that it did! That
is what we propose to do in this chapter, just to prove the existence
of the requisite number of the cubes, fourth powers, etc. We do not
attempt to find the structure of the 4, 9, 19, ..., but just to prove its
existence.
So let us fix k and view the kth powers. Our aim, by Schnirel-
mann’s lemmas below, need be only to produce a g g(kð and an
α αàkð > 0 such that the sum of g(kð kth powers represents at
least the fraction αàkð of all of the integers.
One of the wonderful things about this approach is that it requires
only upper bounds, despite the fact that Waring’s conjecture seems
to require lower bounds, something seemingly totally impossible
for contour integrals to produce. But the adequate upper bounds are
obtained by the so called Weyl sums given below.
So first we turn to our three basic lemmas which will eventually
yield our proof. These are A, the theorem of Dirichlet, B, that of
Schnirelmann, and finally C, the evaluation of the Weyl sums.

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