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V. The Waring Problem
52
The full estimate, then, by the inductive hypothesis is
N 1 N 2+o(1) 1
2 1+o(1) − 1−
|S| dN d 2 k−2 ≤ b 2 k−2 1
b b
d|b d|b
1 o(1)
b
N 2+o(1) − 2 k−2 b .
So we obtain
1
1+o(1) −
S N b 2 k−1 ,
and the induction is complete.
Now we continue as follows:

Lemma 3. Let k> 1 be a fixed integer. There exists a C 1 such that,
for any positive integers N, a, b with (a, b) 1,

N

a 1+o(1) −2 1−k
e n k ≤ C 1 N b .
b
n 1
Our endpoint will be the following:

Theorem. If, for each positive integer s, we write

r s (n) 1,
k k
n +···+n n
1 s
n i ≥0
then there exists g and C such that r g (n) ≤ Cè g/k−1 for all n> 0.

The previously cited notions of Schnirelmann allow deducing, the
full Waring result from this theorem:
There exists a G for which r G (n) > 0 for all n> 0.

To prove our theorem, since

s
1
k
r s (n) eàxm ) eà −nx)dx,
0 m≤n 1/k