ON WARING'S PROBLEM
By LOO-KENG HUA {Cambridge)
[Received 1 February 1938]
THE object of the present paper is to give a proof that Hardy and
Little wood's asymptotic formula for the number of solutions of the
Diophantine equation
is true for s > 2 fc +l. This result is new only for k < 14. For k ^ 14,
Vinogradow's contribution is much better than this. The most interesting particular case is k = 4, for which Estermannf and
Davenport and HeilbronnJ proved that every sufficiently large
integer is a sum of 17 fourth powers, but they gave no asymptotic
formula for the number of solutions.'
More precisely, what I am going to prove is the following more
general theorem:
Let P1(a;),..., Ps(x) be s integral-valued polynomials of the kth degree
and let their first coefficients be positive numbers av..., as respectively.
Let r(N) be the number of solutions of the Diophantine equation
N = P1(x1)+...+P3(xs)
Then, for s ^ 2 +l, we have
(xv^0).
fc
. r(N) = F
T g , - 1 71"(s/lc)
!!/
l=i
where S = 21~ks—z—e and e is an arbitrary small positive number and
<5(N) is defined in one of my previous papers.^
I shall give elsewhere an application of this theorem to prove that
G{P{x)} < 17,
provided that P(x) is a quartic polynomial with coefficient of x4
positive and that there does not exist an-integer q ( > 1) such that
P{x) = P(0) (mod q)
for all x.
The proof of the theorem depends essentially on the following
lemma which seems to have some interest in itself.
t Proc. London Math. Soc. 41 (1938), 127-42.
+ Ibid. 41 (1938), 143-50.
§ Ibid. 43 (1937), 161-82.
200
LOO-KENG HUA
MAIN LEMMA.
kth degree, and
Let P(x) be an integral-valued polynomial of the
• p
/(<*) = 2 exp{27rt P(x)a}.
x=l
Then
J |/(ot)|* da =
o
where (A,/x(A)} lies on a polygonal line with vertices (2V, 2"—H-e)
(v = 1,..., k), and the constants implied by the symbol 0 depend only
on the coefficients of P(x) and e.
An improvement of this lemma for the particular case P(x) = xk
and its application to the additive prime-number theory will appear
elsewhere later.
Proof of the main lemma
Since
log(j|/(«)|"d«)
is a convex function of v,f we only need to prove that
I
J \f(cc)\2"d* = W - + * )
for v = 1, 2,..., k.
(1)
o
Without loss of generality we assume that P(x) is a polynomial
with integer coefficients. In fact, let q be the least common denominator of the coefficients of P(x), then,/by Holder's inequality,
• |/(«)| A da= I 2
2 exp{27rtP(qx+a)<x}\ da.
i 'a=l
a:=0
1
'
" * > - 0m
)1/9
ff exp[2ni{P(qx+a)-P(a)}a]
i JffII""I x=0
X
da,
a = l J I x=0
where P(qx-\-a)—P(a) is a polynomial with integer coefficients.
Then (1) is trivial for v = 1 and well known for v = 2.§ We are
going to prove (1) by induction.
We use the abbreviation
A Q{x) = -{Q{x+y)-Q{x)}.
if
t See, for example, Hardy, Littlewood, and Polya, Inequalities, § 6.12.
§ See, for example, Landau, Vorlesungen iiber Zahlentheorie, Bd. 1, Satz 262,
37. There he deals only with the particular case P(x) = a^. For the general
case see Hua, J. of-Chinese Math. Soc. 1 (1936), 23-61, Lemma 11.
ON WAKING'S PROBLEM
201
Then A Q(x) is a polynomial of the (h—l)th degree in x, provided
that Q(x) is a polynomial of the hth degree. Let 2, denote a summaX
tion with variable x whose number of terms is O(p).
Consider
l/MI2=i 1
= 21
x,
expl2Tri{P(xs+yi)-P(x2)}<x]
Vi
P P
= 2 2 exp{27«y1 A P(a;2)a).
l/l X,
"»
By Schwarz's inequah'ty, we have
l/(«) I4 < P 1 1 exp{2^>1 A P(x2)a}f
< P X 2 2 exp{27rty1 y2 A A P(a;3)a},
y
y
x
V*V\
where -4 <^ JS means ^4 = 0(B). Repeating this process, we obtain,
in general,
exp{27Ti?/1 ...y
A A ... i
^ 1 ...y A ... 4 P ( ^ + 1 H (2)
Vi
V x
"
W
for ft = 1, 2,..., Jfc—1, where * denotes the condition
2/1...2//iA...A
We have, therefore,
j
-" f l - 1 i*exp{2^1...yl,_] A ...
g j/i
J/./-1 xv
Vv-%
vi
By the hypothesis of the induction, the first term on the right of
The second term on the right of (3) equals
-I
1*1 -f
exV[2ni{Vl ...yv_x A
l
U-i
...AP(xv)tfi
-.)} a ] da = p
202
ON WAKING'S PBOBLEM
where R denotes the number of solutions of
yx...yv-x A A...P(xv) = P( Z l )-P(z 2 )+...-P(v-),
Vv—\ l/i
^ . . . ^ . . . A P K ) # 0,
2/i ,
y,,, *„„ < P .
(4)
For given zv..., z2"->, the number of solutions of (4) is
Since d(n) = O(ne), we have
where % denotes the condition P(z1) —P(z2)+...—P(z2^-i) =fc 0. The
lemma is therefore proved.
Proof of the theorem
k
Let p — N^ and
where pr is the greatest root of Pr(x) = N. It is evident that when
N is sufficiently large, pr always exists and p <^j)r <€,p. Then
1 s
r(N)=
Jl Sr(<x)exp{—2iTiotN) dot.
It is sufficient to estimate the part If of the integral corresponding
to the minor arcs, since the remaining part can be treated, without
any difficulty., by the method used in my previous paper.§
By Weyl's theorem!! and the main lemma we have
W <^2yi-2l~*+<*>-2*) f J J \sr(<x)\ da
2*
1
v2
JJ JJ \Sr(a)f da)
The theorem is proved.
In closing, I should like to express my warmest thanks to the
referee for his valuable advice.
f rl(n) denotes number of divisors of >i.
§ l'roc. London Math. Sov. 43 (1937). 161-82.
I Landau, Vorlcsungen iibrr Zttlilnitlicoric, Satz 267.