Download a three squares theorem with almost primes

Bull. London Math. Soc. 37 (2005) 507–513
e 2005 London Mathematical Society
C
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 4k (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
x21 + x22 + x23 , 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) xθ 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 nγ . The number
of such representations exceeds c(log n)−3 (log log n)−2 S(n)n1/2 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.
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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
nγ pn1/ 2 x21 +x22 +x23 =n
x1 =0, p2 |x1
√
pnγ 0<x1 n x22 +x23 =n−x21
p2 |x1 <
nε
n1/2
p2
γ
pn
n(1−γ)/2 .
Our theorem is the analogue of a Lagrange four squares theorem with almost
primes, which was obtained by Brüdern and Fouvry [2] (see also work by HeathBrown and Tolev [4], and by Tolev [11]). They used a four-dimensional sieve method
in combination with Kloosterman’s refinement of the circle method to control the
error term. The sieve part of our arguments is very much the same as that in [2],
and we may therefore direct the uninitiated reader to that paper for a more detailed
discussion of this matter. The circle method, however, is replaced by the theory of
theta-functions and modular forms. The new key ingredient is a uniform estimate
[1] for the number of representations of integers by ternary quadratic forms which
is based on the work of Duke and Schulze-Pillot [3], and on Iwaniec’s bound for
the Fourier coefficients of cusp-forms of half-integral weight [6]. We remark that
further progress towards the Ramanujan–Petersson conjecture in the half-integral
weight case would decrease the number of prime factors in Corollary 1.2.
We adopt the notation used in [2], and we write boldface symbols for threedimensional vectors. Let x ≡ 0 (mod d) denote the simultaneous congruences xi ≡ 0
(mod di ), 1 i 3. Moreover, let |d| = max|di |, and µ(d) = µ(d1 )µ(d2 )µ(d3 ),
where µ is the Möbius function. Let S(n, d) be the formal singular series associated
with the representation of n by the quadratic form d21 x21 + d22 x22 + d23 x23 ; hence
S(n) = S(n, (1, 1, 1)). For n ≡ 3 (mod 8), we write
ω(d) = ω(d, n) = S(n, d)S(n)−1 .
2. Representation of integers by certain quadratic forms
The set to be sieved will be
A = {x ∈ N3 : x21 + x22 + x23 = n}.
In order to apply a sieve method, we need information about the distribution of
the members of A in arithmetic progressions; that is, we need asymptotic formulae
for the cardinality of
Ad = {x ∈ A : x ≡ 0 mod d} = {x ∈ N3 : d21 x21 + d22 x22 + d23 x23 = n}.
For a positive definite ternary quadratic form f (x) = xt Ax, let
r(f, n) := #{x ∈ Z3 : f (x) = n},
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509
and let
o(f ) = #{T ∈ SL3 (Z) : T t AT = A}
be the (finite) number of automorphs of f . As usual, we define the weighted means
−1 1
r(f˜, n)
r(gen f, n) :=
o(f˜)
o(f˜)
˜
˜
f ∈gen f
and
r(spn f, n) :=
f˜∈spn f
f ∈genf
1
o(f˜)
−1 r(f˜, n)
,
o(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, we have:
r(spn f, n) − r(f, n) ε N 45/28 n13/28+ε ,
uniformly in N n1/2 . For squarefree n, we have
r(spn f, n) − r(f, n) ε N n13/28+ε ,
uniformly in N n1/2 .
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 Ad , S(n) and ω(d) be as above. Set
θ0 = 1/252 if n is squarefree, and θ0 = 1/354 otherwise. Define the real numbers
R(n, d) by the equation
ω(d, n) π
S(n)n1/2 + R(n, d).
d1 d2 d3 4
Then, whenever 0 < θ < θ0 , one has
µ2 (d)|R(n, d)| n1/2−ε
|Ad | =
|d|nθ
if ε > 0 is sufficiently small in terms of θ.
Proof. For 2 | d1 d2 d3 , obviously R(n, d) = 0, since ω(d, n) = |Ad | = 0. Therefore we may consider only d with 2 d1 d2 d3 . The forms fd (x) = d21 x21 + d22 x22 + d23 x23
with µ2 (d) = 1, 2 d1 d2 d3 , have, by [8, (102:10)], only one spinor genus per genus,
and thus
2π
S(n, d)n1/2 ,
r(spn fd , n) = r(gen fd , n) =
d1 d2 d3
by a well-known result of Siegel [9]. Also, the previous lemma yields R(n, d) (d1 d2 d3 )45/14 n13/28+ε for odd n and R(n, d) (d1 d2 d3 )2 n13/28+ε for squarefree n,
uniformly in d.
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valentin blomer and jörg brüdern
3. Local computations
To apply the vector sieve, we have to study the behaviour of ω. This can be
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
q
the quaternary case. Let S(q, a) = x=1 e(ax2 /q), let
q
−an
−3
2
2
2
S(q, ad1 )S(q, ad2 )S(q, ad3 )e
A(q, d, n) = q
,
q
a=1
(a,q)=1
and let
χp (n, d) =
∞
A(pk , d, n)
k=0
be the local densities. We write e1 (p) = (p, 1, 1), e2 (p) = (p, p, 1), e3 (p) = (p, p, p)
and ων (p) := ω(eν (p)) for 1 ν 3. Let µ(d)2 = 1. Then, since S(n, d) =
k
k
ν
p χp (n, d) and A(p , d, n) = A(p , eν (p), n) for p d1 d2 d3 , we have
χp (n, eν (p))
=
ων (p).
ω(d) =
χp (n, (1, 1, 1))
ν
ν
p d1 d2 d3
ν1
p d1 d2 d3
ν1
It is clear that ων (2) = 0 for ν 1 and n ≡ 3 (mod 8).
Lemma 3.1. Let p be odd. If p n, then
p − −1
p 1 + np
p
,
,
ω1 (p) =
ω2 (p) =
p + −n
p + −n
p
p
ω3 (p) = 0.
If pθ n with θ 1, then
p−1
1 + −1
1 + p2 fθ (p)
p + p3 fθ (p)
p
p + pfθ (p)
ω1 (p) =
,
ω2 (p) =
,
ω3 (p) =
,
1 + fθ (p)
1 + fθ (p)
1 + fθ (p)
where fθ is defined as follows.
(i) If θ is odd, put fθ (p) = p−1 − p−(θ+1)/2 − p−(θ+3)/2 .
(ii) If θ is even, put
−np−θ
fθ (p) = p−1 − p−(θ+2)/2 +
p−(θ+2)/2 .
p
Proof.
In the case where p n, we have
p χp (n, d) = #{x ∈ (Z/pZ)3 : d21 x21 + d22 x22 + d23 x23 ≡ n (mod p)}
2
(see, for example, [9, Hilfssatz 13]), and the lemma follows easily.
Now let pθ n with θ 1. We see that [9, Hilfssatz 16] yields χp (n, (1, 1, 1)) =
1 + fθ (p). Since A(pk , eν (p), n) = pν A(pk , (1, 1, 1), n) for k 2 (see [2, (2.55)]),
we have
∞
A(pk , (1, 1, 1), n).
χp (n, eν (p)) = 1 + A(p, eν (p), n) + pν
k=2
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 fθ (p).
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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) =
ω1 (p).
p|d
Assume that 3 | n. If d = (d1 , d2 , d3 ) is a triple of squarefree integers, we put
di,j = (di , dj ) for 1 i < j 3. Then the following statements hold.
(i) There exists a function g : N3 −→ R such that for any d, we have
ω(d) = ω(d1 )ω(d2 )ω(d3 )g(d1,2 , d1,3 , d2,3 ).
(ii) There exists an absolute constant C such that for any d, we have
g(d1,2 , d1,3 , d2,3 ) (max di,j )C .
(iii) For any d, we have the inequality
ω(d) ω̃(d1 )ω̃(d2 )ω̃(d3 ),
where ω̃ is the multiplicative function defined on squarefree integers by
2,
if p n,
ω̃(p) =
p2/3 , if p | n.
(iv) For p n, we have ω1 (p) (p + 1)/(p − 1); for p | n, we have
1 + p1 , if p ≡ −1 (mod 4),
ω1 (p) 3,
if p ≡ 1 (mod 4).
Proof. This follows easily from Lemma 3.1 when we observe for part (iii) that
fθ (p) 1/p; thus ω1 (5) = 7/3 < 52/3 , ω2 (p) < p and ω3 (p) < p2 .
As the most important consequences of this lemma, we note that by part (iv) we
always have 0 ω1 (p) < p, and we also see that there exists an absolute constant
L such that whenever 2 w w , one has
−1 log w
ω(p)
L(log log n)2
<
1−
1+
.
(3.1)
p
log w
log w
wpw
4. The vector sieve
For primes p, we define the multiplicative function Ω by
Ω(p) = 3ω1 (p) −
3ω2 (p) ω3 (p)
+
.
p
p2
We have, by Lemma 3.2,


3,
Ω(p) 7,


1,
if p n,
if p | n, p ≡ 1 (mod 4),
if p | n, p ≡ −1 (mod 4),
and hence 0 Ω(p) < p, if n ≡ 3 (mod 24) and 5 n.
(4.1)
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valentin blomer and jörg brüdern
Furthermore, we have
µ(d)
d
p|d1 d2 d3 =⇒p<z0
where
W (z0 ) =
ω(d)
= W (z0 ),
d1 d2 d3
p<z0
1−
Ω(p)
.
p
We can now formulate a fundamental lemma for the vector sieve.
Lemma 4.1. Let z0 2. Let l be an integer triple with µ2 (l) = 1 such that all
prime factors of l1 , l2 and l3 exceed z0 . Let
S(Al , z0 ) = #{x ∈ Al : p | xi =⇒ p z0 }.
Then, for D0 z02 , ∆ 1, one has
S(Al , z0 )
π ω(l)
S(n)n1/2
= W (z0 ) + O(H(n)4 ∆−1/2 (log D0 )19 + (log n)L ∆c e−s0 )
4
l
l
l
1
2
3
µ2 (d)|R(n, (d1 l1 , d2 l2 , d3 l3 ))|
+O
.
|d|D0
p|d1 d2 d3 =⇒p<z0
The implied constants are absolute, c is an absolute constant, L is the same constant
as in (3.1), and we have written
log D0
and
H(n) =
(1 + p−1/6 ).
s0 =
log z0
p|n
An important ingredient for the lower bound is the following inequality, which is
the analogue of [2, Lemma 13], and which is used with Λ± = λ± ∗ 1, where λ± are
Rosser’s weights (see [5]).
+
0
0
Lemma 4.2. For real numbers Λ−
i , Λi and Λi (i = 1, 2, 3) with Λi ∈ {0, 1} and
+
0
Λi Λi , we have
Λ−
i
+ +
+ − +
+ + −
+ + +
Λ01 Λ02 Λ03 Λ−
1 Λ2 Λ3 + Λ1 Λ2 Λ3 + Λ1 Λ2 Λ3 − 2Λ1 Λ2 Λ3 .
The proof of Lemma 4.2 is the same as that given in [2]. The proof of Lemma 4.1
follows along the lines of the proof of the analogous proposition in [2, pp. 87–92],
and we have only to change some exponents, and to use the results from Lemma 3.3.
The additional log-power in O(H(n)4 ∆−1/2 (log D0 )19 + (log n)L ∆c e−s0 ) is due to
the occurrence of a factor (log log n)2 in (3.1).
We proceed to prove the theorem. By (4.1), we have
W (z0 ) max (log z0 )−7 , (log z0 )−3 (log log n)−2 .
We can now copy the proof of [2, Theorem 1], with some obvious changes of
exponents. We use Lemma 4.1 for a preliminary sieving with z0 = (log n)30 , say,
and appeal once again to Lemma 3.2, obtaining an inequality similar to [2, (3.28)].
A numerical calculation finally yields 3f (s) − 2F (s) > 0 for s s0 = 2.948, where
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
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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.
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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
[email protected]
Jörg Brüdern
Institut für Algebra und Zahlentheorie
Universität Stuttgart
Pfaffenwaldring 57
D-70569 Stuttgart
Germany
[email protected]