a three squares theorem with almost primes - Mathematisches Institut

Bull. London Math. Soc. 37 (2005) 507–513
C❡ 2005 London Mathematical Society
doi:10.1112/S0024609305004480
A THREE SQUARES THEOREM WITH ALMOST PRIMES
VALENTIN BLOMER and JÖRG BRÜDERN
Abstract
As an application of the vector sieve and uniform estimates on the Fourier coefficients of cusp
forms of half-integral weight, it is shown that any sufficiently large number n ≡ 3 (mod 24) with
5 ∤ n is expressible as a sum of three squares of integers having at most 521 prime factors.
1. Introduction
Gauß proved that all integers n not of the form 4 k (8m + 7) can be written as a
sum of three squares; thus all n not excluded by obvious congruence conditions
have this property. Since there is only one class in the genus of the quadratic form
x 2 1 + x 2 2 + x 2 3, the number of such representations can be given explicitly (see [9]). It
is conjectured that the three squares theorem still holds if we restrict the variables
to prime numbers, as long as its validity is not precluded by local conditions. Thus
we assume that n ≡ 3 (mod 24) and 5 ∤ n. One step in this direction is the following
result of Liu and Zhan [7]: If E(x) is the set of positive integers less than or equal
to x that satisfy these congruence conditions, but cannot be represented as a sum
of three squares of primes, then E(x) ≪ x θ for any θ>47/50; thus the set of
exceptions is thin.
In this paper we choose another approach to the conjecture mentioned above. Let
S(n) be the formal singular series associated with the representation of n as a sum
of three squares. Then, by Siegel’s [9, Hilfssätze 12 and 16] and Siegel’s theorem
[10], we have
S(n) ≫ (log log n) −1 L(1,χ −4n ) ≫ ε n −ε
for n ≡ 3 (mod 8) and for all ε>0. We shall prove the following theorem.
Theorem 1.1. Let n ≡ 3 (mod 24), 5 ∤ n, be sufficiently large, and let
γ =10/7429 if n is squarefree and γ =5/5218 otherwise. Then n is the sum of
three squares of integers with all their prime factors greater than n γ . The number
of such representations exceeds c(log n) −3 (log log n) −2 S(n)n 1/2 for some positive
constant c.
The asymptotic behaviour of the number of representations is close to what one
would expect. Since we have to appeal to Siegel’s theorem, the result is not effective.
The following corollary is immediate.
Corollary 1.2. Every large n ≡ 3 (mod 24) not divisible by 5 is the sum of
three squares of integers having at most 521 prime factors; if n is squarefree, it is
the sum of three squares of integers having at most 371 prime factors.
Received 29 August 2002, revised 19 August 2004.
2000 Mathematics Subject Classification 11P05, 11N36, 11N75, 11E25.

508 valentin blomer and jörg brüdern
By working along the lines proposed by Tolev [11], it is possible to reduce the
number of prime factors in this corollary.
Corollary 1.3. Every sufficiently large n ≡ 3 (mod 24) not divisible by 5 is
the sum of three squares of squarefree numbers.
This is also clear, since only a small proportion of the representations obtained
by the theorem can have non-squarefree variables:
∑ ∑
1 ≪ ∑ ∑ ∑
1
n γ pn 1/ 2
x 2 1 +x2 2 +x2 3 =n
x 1≠0, p 2 |x 1
pn γ
≪ n ε ∑
0

a three squares theorem with almost primes 509
and let
o(f) =#{T ∈ SL 3 (Z) :T t AT = A}
be the (finite) number of automorphs of f. As usual, we define the weighted means
( ∑
) −1
1 ∑ r(
r(gen f,n) :=
˜f,n)
o( ˜f)
o( ˜f)
˜f∈gen f
˜f∈genf
and
( ∑
) −1
1 ∑ r(
r(spn f,n) :=
˜f,n)
o( ˜f)
o( ˜f) ,
˜f∈spn f
˜f∈spn f
where the summations are taken over a set of representatives of all classes in the
genus and the spinor genus of f, respectively (see [8], Section 102). By a well-known
result of Siegel [9], r(gen f,n) is the product of all local densities. It turns out that
r(spn f,n) is a good approximation to r(f,n).
Lemma 2.1. Let f be a positive definite ternary quadratic form of level N.
Then, for any ε>0 and odd n, wehave:
r(spn f,n) − r(f,n) ≪ ε N 45/28 n 13/28+ε ,
uniformly in N n 1/2 . For squarefree n, wehave
r(spn f,n) − r(f,n) ≪ ε Nn 13/28+ε ,
uniformly in N n 1/2 .
Lemma 2.1 is part of [1, Theorem 1], and implies the following result, which is
the analogue of [2, Theorem 3].
Lemma 2.2. Let n ≡ 3 (mod 8), and let A d , S(n) and ω(d) be as above. Set
θ 0 =1/252 if n is squarefree, and θ 0 =1/354 otherwise. Define the real numbers
R(n, d) by the equation
|A d | = ω(d,n) π
d 1 d 2 d 3 4 S(n)n1/2 + R(n, d).
Then, whenever 0

510 valentin blomer and jörg brüdern
3. Local computations
To apply the vector sieve, we have to study the behaviour of ω. Thiscanbe
done similarly as in [2, pp. 75–78], so we content ourselves with a rough outline. It
turns out that in the ternary case, ω behaves somewhat more irregularly than in
the quaternary case. Let S(q, a) = ∑ q
x=1 e(ax2 /q), let
q∑
( ) −an
A(q, d,n)=q −3 S(q, ad 2 1)S(q, ad 2 2)S(q, ad 2 3)e ,
q
and let
a=1
(a,q)=1
χ p (n, d) =
∞∑
A(p k , d,n)
k=0
be the local densities. We write e 1 (p) =(p, 1, 1), e 2 (p) =(p, p, 1), e 3 (p) =(p, p, p)
∏
and ω ν (p) := ω(e ν (p)) for 1 ν 3. Let µ(d) 2 = 1. Then, since S(n, d) =
p χ p(n, d) andA(p k , d,n)=A(p k , e ν (p),n)forp ν ‖ d 1 d 2 d 3 ,wehave
ω(d) =
∏
p ν ‖d 1d 2d 3
ν1
χ p (n, e ν (p))
χ p (n, (1, 1, 1)) =
It is clear that ω ν (2) = 0 for ν 1andn ≡ 3 (mod 8).
∏
p ν ‖d 1d 2d 3
ν1
ω ν (p).
Lemma 3.1. Let p be odd. If p ∤ n, then
ω 1 (p) = p − ( )
−1
p
p + ( −n
), ω 2 (p) = p( 1+ ( ))
n
p
p
p + ( )
−n
, ω 3 (p) =0.
p
If p θ ‖ n with θ 1, then
ω 1 (p) = 1+( −1
p
) p−1
p
1+f θ (p)
+ pf θ (p)
, ω 2 (p) = 1+p2 f θ (p)
1+f θ (p) , ω 3(p) = p + p3 f θ (p)
1+f θ (p) ,
where f θ is defined as follows.
(i) If θ is odd, put f θ (p) =p −1 − p −(θ+1)/2 − p −(θ+3)/2 .
(ii) If θ is even, put
( ) −np
f θ (p) =p −1 − p −(θ+2)/2 −θ
+
p −(θ+2)/2 .
p
Proof. In the case where p ∤ n, wehave
p 2 χ p (n, d) =#{x ∈ (Z/pZ) 3 : d 2 1x 2 1 + d 2 2x 2 2 + d 2 3x 2 3 ≡ n (mod p)}
(see, for example, [9, Hilfssatz 13]), and the lemma follows easily.
Now let p θ ‖ n with θ 1. We see that [9, Hilfssatz 16] yields χ p (n, (1, 1, 1)) =
1+f θ (p). Since A(p k , e ν (p),n)=p ν A(p k , (1, 1, 1),n)fork 2(see[2, (2.55)]),
we have
χ p (n, e ν (p)) = 1 + A(p, e ν (p),n)+p ν
∞
∑
k=2
A(p k , (1, 1, 1),n).
This can easily be calculated by observing that A(p, (1, 1, 1),n) = 0, whence the
sum on the right-hand side is just f θ (p).

a three squares theorem with almost primes 511
The following lemma summarizes the properties of ω(d) that we need for the
proof of the theorem.
Lemma 3.2.
For squarefree d ∈ N, let
ω(d, n) =ω(d) = ∏ p|d
ω 1 (p).
Assume that 3 | n. Ifd =(d 1 ,d 2 ,d 3 ) is a triple of squarefree integers, we put
d i,j =(d i ,d j ) for 1 i

512 valentin blomer and jörg brüdern
Furthermore, we have
∑
µ(d) ω(d) = W (z 0 ),
d 1 d 2 d 3
d
p|d 1d 2d 3=⇒p

a three squares theorem with almost primes 513
f and F denote the classical functions of the linear sieve; by Lemma 2.2, we can
sieve all primes not exceeding n (θ0−ε)/s0 .
References
1. V. Blomer, ‘Uniform bounds for Fourier coefficients of theta series with arithmetic
applications’, Acta Arith. 114 (2004) 1–21.
2. J. Brüdern and E. Fouvry, ‘Lagrange’s four squares theorem with almost prime variables’,
J. Reine Angew. Math. 454 (1994) 59–96.
3. W. Duke and R. Schulze-Pillot, ‘Representation of integers by positive ternary quadratic
forms and equidistribution of lattice points on ellipsoids’, Invent. Math. 99 (1990) 49–57.
4. D. R. Heath-Brown and D. I. Tolev, ‘Lagrange’s four squares theorem with one prime and
three almost-prime variables’, J. Reine Angew. Math. 558 (2003) 159–224.
5. H. Iwaniec, ‘Rosser’s sieve’, Acta Arith. 36 (1980) 171–202.
6. H. Iwaniec, ‘Fourier coefficients of modular forms of half-integral weight’, Invent. Math. 87
(1987) 385–401.
7. J. Liu and T. Zhan, ‘Distribution of integers that are sums of three squares of primes’, Acta
Arith. 98 (2001) 207–228.
8. O. T. O’Meara, Introduction to quadratic forms (Springer, 1973).
9. C. L. Siegel, ‘Über die analytische Theorie quadratischer Formen I’, Ann. Math. 36 (1935)
527–606.
10. C. L. Siegel, ‘Über die Classenzahl quadratischer Zahlkörper’, Acta Arith. 1 (1935) 83–86.
11. D. I. Tolev, ‘Lagrange’s four squares theorem with variables of special type’, Proceedings of
the session in analytic number theory and Diophantine equations held in Bonn, Germany,
January–June, 2002, Bonner Mathematische Schriften 360 (ed. D. R. Heath-Brown, et al.,
Mathematisches Institut., Univ. Bonn, Bonn, 2003).
Valentin Blomer
Department of Mathematics
100 St Georges Street
Toronto, Ontario
Canada M5S 3G3
blomer@uni-math.gwgd.de
Jörg Brüdern
Institut für Algebra und Zahlentheorie
Universität Stuttgart
Pfaffenwaldring 57
D-70569 Stuttgart
Germany
bruedern@mathematik.uni-stuttgart.de