Download On the number of primitive representations of integers as sums of

Ramanujan J (2007) 13:7–25
DOI 10.1007/s11139-006-0240-6
On the number of primitive representations of integers
as sums of squares
Shaun Cooper · Michael Hirschhorn
Dedicated to Richard Askey on the occasion of his 70th birthday.
Received: 12 April 2002 / Accepted: 8 October 2002
C Springer Science + Business Media, LLC 2007
Abstract Formulas for the number of primitive representations of any integer n as a
sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12.
The formulas have a similar structure and are striking for their simplicity.
Keywords Sum of squares . Möbius inversion . Generating function . Divisor sum
2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15,
33E05
1 Introduction
Let rk (n) denote the number of representations of n as a sum of k squares. That is, rk (n)
is the number of solutions in integers, counting permutations and sign changes, of
x12 + x22 + · · · + xk2 = n.
(1.1)
A solution of (1.1) is called primitive if g.c.d.(x1 , x2 , . . . , xn ) = 1. Here we use the
convention that any integer is a divisor of zero. Let us denote the number of primitive
p
representations of n as a sum of k squares by rk (n).
S. Cooper
Institute of Information and Mathematical Sciences, Massey University,
Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand
e-mail: [email protected]
M. Hirschhorn
School of Mathematics, University of New South Wales, Sydney 2052, Australia
e-mail: [email protected]
Springer
8
S. Cooper, M. Hirschhorn
For example, consider the representations of 225 as a sum of three squares:
Number of representations
Primitive representations
23 × 3! = 48
23 × 3! = 48
2×3=6
22 × 3! = 24
23 × 3 = 24
(14, 5, 2)
(11, 10, 2)
(15, 0, 0)
(12, 9, 0)
(10, 10, 5)
Non-primitive representations
Thus
p
r3 (225) = 48 + 48 = 96,
r3 (225) = 48 + 48 + 6 + 24 + 24 = 150.
p
The quantities rk (n) and rk (n) are clearly related by [18, p. 6]
p n rk (n) =
rk
.
d2
d 2 |n
By the Möbius inversion formula, we also have
n n
p
rk (n) = rk (n) −
+
−
rk
rk
p12
p12 p22
p 2 |n
p 2 , p 2 |n
1
+ · · · + (−1) j
1
2
n
p12 · · · p 2j
rk
p12 ,... , p 2j |n
,
p12 , p22 , p32 |n
(1.2)
rk
n
2 2 2
p1 p2 p3
(1.3)
where p1 , . . . , p j are the distinct primes whose squares divide n. Observe that if m
is squarefree, then
p
rk (m) = rk (m).
For |q| < 1, let
φ(q) =
∞
qn
2
n=−∞
and define
(a; q)∞ =
∞
(1 − aq j−1 ).
j=1
Then the generating function for rk (n) is
φ(q)k =
∞
n=0
Springer
rk (n)q n .
(1.4)
On the number of primitive representations of integers as sums of squares
9
Throughout this article, n will always be a positive integer, and we will denote its
prime factorization by
n = 2 λ2
pλ p ,
(1.5)
p
where the product is taken over all odd primes p which divide n. The Legendre symbol
is defined, for prime values of p, by
⎧
⎪
⎨ 1 if a is a quadratic residue (mod p)
a
= −1 if a is a quadratic nonresidue (mod p)
⎪
p
⎩
0 if a ≡ 0 (mod p).
For odd values of n, the Jacobi symbol is defined by
⎧
1
if n = 1
⎪
⎪
⎨
a
= a λp
⎪
n
if n > 1,
⎪
⎩
p
p
where p and λ p are as in the prime factorization (1.5).
The main results in this article are:
Theorem 1. Let the prime factorization of n be given by (1.5). Then
p
r2 (n) = c2 (n)
1 + (−1)( p−1)/2 ,
(1.6)
p
1
1+
= c4 (n)n
,
p
p
(−1)( p−1)/2
p
2
r6 (n) = c6 (n)n
1+
,
p2
p
1
p
3
r8 (n) = c8 (n)n
1+ 3 ,
p
p
p
r4 (n)
where
0 if n ≡ 0 (mod 4)
c2 (n) =
4 if n ≡ 0 (mod 4),
⎧
8 if n ≡ 1 (mod 2)
⎪
⎪
⎪
⎨12 if n ≡ 2 (mod 4)
c4 (n) =
⎪
4 if n ≡ 4 (mod 8)
⎪
⎪
⎩
0 if n ≡ 0 (mod 8),
⎧
12 if n ≡ 1
(mod 4)
⎪
⎪
⎪
⎨20 if n ≡ 3
(mod 4)
c6 (n) =
⎪
15 if n ≡ 0, 2, or 4 (mod 8)
⎪
⎪
⎩
17 if n ≡ 6
(mod 8),
(1.7)
(1.8)
(1.9)
(1.10)
(1.11)
(1.12)
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10
S. Cooper, M. Hirschhorn
⎧
16
⎪
⎪
⎪
⎪
⎪
⎨14
c8 (n) = 35
⎪
⎪
⎪
2
⎪
⎪
⎩
18
if n ≡ 1
(mod 2)
if n ≡ 2
(mod 4)
if n ≡ 4
(mod 8)
if n ≡ 0
(mod 8).
(1.13)
(1.14)
If at least one of the exponents λ p is odd for some prime p ≡ 3(mod 4), then
p
r10 (n) = c10 (n)n 4
where
⎧
68
⎪
⎪
⎪
⎪
⎪
5
⎪
⎪
⎪
⎪
12
⎪
⎨
c10 (n) = 51
⎪
⎪
⎪
⎪4
⎪
⎪
⎪
⎪
⎪
257
⎪
⎩
20
(−1)( p−1)/2
1+
,
p4
p
if n ≡ 1
(mod 4)
if n ≡ 3
(mod 4)
if n ≡ 0, 4, or 6
if n ≡ 2
(mod 8)
(1.15)
(1.16)
(mod 8).
Furthermore, if n ≡ 0, 2, or 6 (mod 8), then
p
1+
(1.17)
⎧
33
⎪
⎪
if n ≡ 2 (mod 4)
⎨
4
c12 (n) =
⎪
⎪
⎩ 495 if n ≡ 0 (mod 8).
64
(1.18)
p
where
1
p5
,
r12 (n) = c12 (n)n 5
Theorem 2. Let the prime factorization of n be given by (1.5). Let m denote the
squarefree part of n, let k ≥ 1, and let ( ap ) denote the Legendre symbol. Then
−m n 1/2 p
p
p
1−
r3 (n) = c3 (n)r3 (m) 1/2
,
(1.19)
m
p
p| n
m
p
r5 (n)
=
p
c5 (n)r5 (m)
m n 3/2 p
1− 2 ,
m 3/2 p| n
p
m
Springer
(1.20)
On the number of primitive representations of integers as sums of squares
p
r7 (n)
=
p
c7 (n)r7 (m)
11
−m n 5/2 p
1−
,
m 5/2 p| n
p3
(1.21)
m
where
1 if n ≡ 0 (mod 4)
c3 (n) =
0 if n ≡ 0 (mod 4),
⎧
1 if n ≡ 0 (mod 4)
⎪
⎪
⎪
⎪
1
⎪
⎪
⎪
if n = 4k (4 j + 2)
⎪
⎪
2
⎪
⎪
⎨
1
c5 (n) =
if n = 4k (4 j + 3)
⎪2
⎪
⎪
⎪
⎪
1 if n = 4k (8 j + 1)
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 5 if n = 4k (8 j + 5)
7
⎧
1
if n ≡ 0 (mod 4)
⎪
⎪
⎪
⎪
⎪
5
⎪
⎪
if n = 4k (4 j + 1)
⎪
⎪
4
⎪
⎪
⎨
c7 (n) = 5
if n = 4k (4 j + 2)
⎪
4
⎪
⎪
⎪
⎪
1
if n = 4k (8 j + 3)
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 35 if n = 4k (8 j + 7).
37
(1.22)
(1.23)
(1.24)
Furthermore,
p
r9 (n)
=
p
r9 (m)
m n 7/2 p
1− 4
m 7/2 p| n
p
if m ≡ 5
(mod 8),
(1.25)
if m ≡ 7
(mod 8).
(1.26)
m
p
r11 (n)
=
p
r11 (m)
−m n 9/2 p
1−
m 9/2 p| n
p5
m
For completeness, we also state the results for r2k+1 (m) for squarefree values of m,
for 1 ≤ k ≤ 5.
Theorem 3. Let m be squarefree and let ( mj ) denote the Jacobi symbol. Then
r3 (1) = 6,
r3 (2) = 12,
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12
S. Cooper, M. Hirschhorn
⎧ j
(m−1)/2
⎪
j
⎪
if m ≡ 1 (mod 8) and m = 1
(−1)
12
⎪
⎪
j=1
⎪
m
⎪
⎪
⎪
⎪
⎪
(m−1)/2 j ⎪
⎪
⎪
⎨8
if m ≡ 3 (mod 8)
j=1
m
r3 (m) =
⎪
⎪
⎪
(m−1)/2
⎪
j
⎪
j
⎪
(−1)
−12
if m ≡ 5 (mod 8)
⎪
⎪
j=1
m
⎪
⎪
⎪
⎪
⎪
⎩
0
if m ≡ 7 (mod 8),
(1.27)
⎧ m/2
j
j
m/8
⎪
⎪
−24
if m ≡ 1 (mod 4)
24
⎪
⎪
j=1
j=3m/8+1
m
m
⎨
r3 (2m) =
(1.28)
and m = 1
⎪
⎪
3m/8
j
⎪
⎪
⎩24
if m ≡ 3 (mod 4).
j=m/8+1 m
r5 (1) = 10,
r5 (2) = 40,
⎧
(m−1)/2 j ⎪
⎪
j if m ≡ 1 (mod 8) and m = 1
⎪−80
j=1
⎪
m
⎪
⎪
⎪
⎪
⎪
⎪
(m−1)/2
⎪
j
⎪
j
⎪
−80
(−1)
j if m ≡ 3 (mod 8)
⎪
⎪
j=1
m
⎨
r5 (m) =
(m−1)/2 j ⎪
⎪
⎪
−112
j if m ≡ 5 (mod 8)
⎪
⎪
j=1
⎪
m
⎪
⎪
⎪
⎪
⎪
⎪
(m−1)/2
j
⎪
j
⎪
⎪
80
(−1)
j if m ≡ 7 (mod 8),
⎩
j=1
m
r5 (2m) = 80m
m−1
j=1
j≡1 (mod 4)
2
j
+ 80(−1)(m−1)/2
j
m
m−1
j=1
j≡3 (mod 4)
2
j
(m − j)
j
m
if m is odd and m > 1.
r7 (1) = 14,
r7 (2) = 84,
Springer
(1.29)
(1.30)
On the number of primitive representations of integers as sums of squares
⎧ j
(m−1)/2
j
⎪
⎪
(m 2 − 4 j 2 )
(−1)
28
⎪
j=1
⎪
m
⎪
⎪
⎪
⎪280 (m−1)/2 j
⎪
⎪
(m 2 − 6 j 2 )
⎪
j=1
⎪
m
⎨ 3
r7 (m) =
(m−1)/2
j
⎪
j
⎪
−28
(−1)
(m 2 − 4 j 2 )
⎪
⎪
j=1
⎪
m
⎪
⎪
⎪
⎪
⎪
(m−1)/2 j ⎪
⎪
⎩−592
j2
j=1
m
r7 (2m) = 56
m−1
j=1
j≡1 (mod4)
13
if m ≡ 1 (mod 8) and m = 1
if m ≡ 3 (mod 8)
if m ≡ 5 (mod 8)
(1.31)
if m ≡ 7 (mod 8),
2
j
(2m 2 + 2m j − j 2 )
j
m
+ 112(−1)(m−1)/2
m−1
j=1
j≡3 (mod 4)
2
j
m(m − j) if m is odd and m > 1.
j
m
(1.32)
r9 (m) =
r11 (m) =
(m−1)/2 j
4320 j 2 (4 j − 3m),
17
m
j=1
(m−1)/2 j
31680 j 3 ( j − m),
31
m
j=1
if m ≡ 5 (mod 8).
if m ≡ 7 (mod 8).
(1.33)
(1.34)
The purpose of this article is to prove Theorems 1 and 2, which we will do in
Sections 2 and 3, respectively.
Equations (1.7) and (1.11), in the case when n is odd, were stated by Eisenstein
[10]. Equations (1.27) and (1.28) were obtained by Dirichlet [8, p. 101], [9]. He obtained these formulas by computing the number of properly primitive classes of binary
quadratic forms of negative determinant, and then applying a theorem of Gauss [7, pp.
262, 265], [13, Section 291], which connects this class number with r3 (n). The equations in (1.27) were restated by Eisenstein [11], who attributed them to Dirichlet [9].
Formulas equivalent to (1.20), (1.21), (1.23), (1.24), (1.29), (1.30), (1.31) and (1.32)
were stated without proof by Smith [47]; see also [7, pp. 308–309]. Equations (1.29),
(1.30), (1.31) and (1.32) had been stated earlier by Eisentein [11, 12].
The Paris Academy of Sciences, noting Eisenstein’s work [11] but apparently unaware of Smith’s work [47], proposed as its Grand Prix des Sciences Mathématiques
competition for 1882 the problem of completely determining the value of r5 (n). The
prize was awarded jointly to Smith [49] and Minkowski [33], who both gave formulas
as well as proofs. An interesting account of this competition and the controversy
surrounding it has been given by Serre [45]; also see [7, p. 312].
The Eqs. (1.33) and (1.34) were stated as conjectures in [4, 5].1
1
Note added in proof: These conjectures have recently been proved by S. Gun and B. Ramakrishnan, On
special values of certain L-functions, The Ramanujan Journal, to appear.
Springer
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S. Cooper, M. Hirschhorn
2 Proof of the formulas for sums of an even number of squares
We begin with the generating functions
Lemma 1.
φ(q)2 = 1 − 4
∞
(−1) j q 2 j−1
1 − q 2 j−1
j=1
φ(q)4 = 1 + 8
∞
j=1
φ(q)6 = 1 + 4
(2.1)
jq j
,
1 + (−q) j
(2.2)
∞
∞
(−1) j (2 j − 1)2 q 2 j−1
j 2q j
+
16
,
2
j−1
1−q
1 + q2 j
j=1
j=1
φ(q)8 = 1 + 16
∞
j=1
φ(q)10 = 1 −
,
j 3q j
,
1 − (−q) j
(2.3)
(2.4)
∞
∞
(−1) j (2 j − 1)4 q 2 j−1
j 4q j
4
64 +
5 j=1
1 − q 2 j−1
5 j=1 1 + q 2 j
+
φ(q)12 = 1 + 8
32 (q 2 ; q 2 )14
∞
,
q
5 (−q; −q)4∞
∞
j=1
j 5q j
+ 16q(q 2 ; q 2 )12
∞.
1 + (−q) j
(2.5)
(2.6)
Proof: Equations (2.1)–(2.4) are due to Jacobi [25, Sections 40 and 42], and Eqs.
(2.5) and (2.6) are due to Glaisher [14]. Glaisher [15, 16] also found similar formulas
for φ(q)14 , φ(q)16 and φ(q)18 . A general formula for φ(q)2k was stated by Ramanujan
[41] and proved by Mordell [34]. Proofs of (2.1)–(2.6) using theta functions were
given by Rademacher [40] and Grosswald [18]. An elementary proof of (2.1)–(2.6),
as well as Ramanujan’s general formula for φ(q)2k , was given in [3].
Lemma 2. Let d j (n) be the number of divisors of n of the form 4i + j. Then
r2 (n) = 4{d1 (n) − d3 (n)},
r4 (n) = 8
d
d|n,4/|d
=8
d|n
r6 (n) = 4
de=n
d odd
Springer
d − 32
d,
(2.7)
(2.8)
d| n4
(−1)(d−1)/2 (4e2 − d 2 ),
(2.9)
On the number of primitive representations of integers as sums of squares
r8 (n) = 16(−1)n
= 16
(−1)d d 3
d|n
d − 32
3
15
d 3 + 256
d| n2
d|n
d 3.
(2.10)
d| n4
Let the prime factorization of n be given by (1.5). If at least one of the exponents λ p
is odd for some prime p ≡ 3 (mod 4), then
r10 (n) =
4
(−1)(d−1)/2 (d 4 + 16e4 ).
5 de=n
(2.11)
d odd
If n is even, then
r12 (n) = 8
d 5 − 512
d5
24
(10 × 25λ2 + 21)
d5
31
d|n
=8
(2.12)
d| n4
d|n
=
(2.13)
d odd
(−1)
d .
d+e−1 5
(2.14)
de=n
Proof: Formulas (2.7)–(2.10) follow by comparing the coefficients of q n in (2.1)–
(2.4), respectively. These were probably known to Jacobi, although he doesn’t seem
to have explicitly stated the results in full. See, for example [26, 27]. Eisenstein
[10], in 1847, referred to “den bekannten Sätzen für 2 and 4 Quadrate” (the well
known theorems for two and four squares) before giving results for the number of
representations of an odd integer as a sum of six or eight squares, and the number
4n + 3 as a sum of ten squares, but he did not explicitly state the two or four squares
results (2.7), (2.8). Nor did he state the six and eight squares results (2.9), (2.10) for
even integers. A complete statement of (2.8)–(2.10) was given in 1865 by Smith [48].
Smith [46] had previously given (2.7) in 1861.
Let
q
∞
(q 2 ; q 2 )14
∞
=
χ4 (n)q n .
(−q; −q)4∞
n=1
(2.15)
Glaisher [15, p. 178, footnote], [17], showed that χ4 (n) = 0 whenever at least one of
the exponents λ p is odd for some prime p ≡ 3 (mod 4). Equation (2.11) then follows
by equating the coefficients of q n on both sides of (2.5). The product in (2.15) can also
be expanded explicitly by one of the Macdonald identities for BC2 . See [6, p. 142],
[32, p. 138, formula BCl (d), with l = 2]. The formula (2.11), in the special case n ≡ 3
(mod 4), was stated without proof by Eisenstein [10]. For completeness, we mention
Springer
16
S. Cooper, M. Hirschhorn
the formula
4 4(λ2 +1) λ p ( p−1)/2 (d−1)/2 4 r10 (n) =
+ (−1)
(−1)
d 2
d|n
5
p
d odd
8
64 2 2
+ n 2r2 (n) −
s t ,
5
5 s 2 +t 2 =n
(2.16)
which holds for all n. This was stated without proof by Liouville [29], and rediscovered
and proved by Glaisher [14, 16]. Observe that by (2.7), Liouville’s formula (2.16)
reduces to (2.11) if at least one of the exponents λ p is odd for some prime p ≡ 3
(mod 4).
For sums of twelve squares, observe that if n is even, then the coefficient of q n
in the infinite product on the right hand side of (2.6) is zero. Therefore (2.12)–(2.14)
follow by comparing coefficients of q n in (2.6).
Lemma 3. Let the prime factorization of n be given by (1.5). Then
r2 (n) = 4
(λ p + 1)
p≡1 (mod 4)
⎧ ⎪
⎨ 8
p
r4 (n) =
⎪
⎩24
r6 (n) = 4 2
p
1 + (−1)λ p
,
2
p≡3 (mod 4)
p λ p +1 −1
,
p−1
if λ2 = 0,
p λ p +1 −1
,
p−1
if λ2 ≥ 1,
2λ2 +2
−
p
r8 (n) =
(−1)
λ p ( p−1)/2
(2.17)
(2.18)
p 2λ p +2 − (−1)(λ p +1)( p−1)/2
,
p 2 − (−1)( p−1)/2
p
p 3λ p +3 − 1
16 3λ2 +3
− 15|
|2
.
7
p3 − 1
p
(2.19)
(2.20)
If at least one of the exponents λ p is odd for some prime p ≡ 3 (mod 4), then
4(λ p +1)
p
− (−1)(λ p +1)( p−1)/2
4 4(λ2 +1) λ p ( p−1)/2
r10 (n) =
2
+ (−1)
,
(2.21)
5
p 4 − (−1)( p−1)/2
p
p
If n is even, then
r12 (n) =
p 5(λ p +1) − 1
24
(10 × 25λ2 + 21)
.
31
p5 − 1
p
(2.22)
Proof: In Eqs. (2.7)–(2.14), express the divisors of n in terms of their prime factorizations, and sum the resulting geometric series.
Formula (2.17) was given in Section 182 of Disquisitiones Arithmeticae by C. F.
Gauss in 1801, [13, footnote on p. 149]. It was also given by Jacobi [27] (who stated
it for even values of n), Ramanujan [42, p. 281, Eqs. 271 & 272], and Hardy and
Springer
On the number of primitive representations of integers as sums of squares
17
Wright [21, Theorem 278]. Formulas (2.18)–(2.20) were given by Ramanujan [42,
pp. 305–307, Eqs. 387, 388, 393, 394, 399 & 400]. We have not been able to find
(2.21) or (2.22) in the literature.
Proof of Theorem 1 in the case k = 6
The results in Theorem 1 follow by applying Eq. (1.3) to Eqs. (2.17)–(2.22). We shall
p
give complete details for r6 (n). The other formulas are proved similarly.
Let
f (2, λ, ε) = 22λ+2 − g( p, λ) =
p 2λ+2 − (−1)(λ+1)( p−1)/2
p 2 − (−1)( p−1)/2
Observe that
r6 (n) = 4 f (2, λ2 , )
g( p, λ p )
p
where
=
(−1)λ p ( p−1)/2 ,
p
and
g( p, −1) = 0.
(2.23)
We consider two cases.
Case 1. n ≡ 0 (mod 4)
Using (1.3) we get
p
r6 (n)
= r6 (n) −
r6
p12 |n
⎡
= r6 (n)⎣1 −
= 4 f (2, λ2 , )
×⎣1 −
+
r6 n/ p12
p12 |n
⎡
n
p12
r6
p12 , p22 |n
r6 (n)
+
n
p12 p22
− · · · + (−1)
r6 n/ p12 p22
p12 , p22 |n
r6 (n)
j
r6
p12 ,... , p 2j |n
n
p12 . . . p 2j
⎤
r6 n/ p12 . . . p 2j
⎦
− · · · + (−1) j
r6 (n)
2
2
p1 ,... , p j |n
g( p, λ p )
p|n
g( p1 , λ p − 2) g( p2 , λ p − 2)
g( p1 , λ p − 2)
1
1
2
+
g(
p
,
λ
)
g( p1 , λ p1 )
g( p2 , λ p2 )
1
p1
2
2 2
p1 |n
p1 , p2 |n
Springer
18
S. Cooper, M. Hirschhorn
− · · · + (−1)
j
p12 ,... , p 2j |n
= 4 f (2, λ2 , )
⎤
g( p j , λ p j − 2)
g( p1 , λ p1 − 2)
⎦
···
g( p1 , λ p1 )
g( p j , λ p j )
g( p, λ p )
1−
p 2 |n
p|n
g( p, λ p − 2)
.
g( p, λ p )
Using (2.23), the second product can be extended to all odd primes p, giving
p
r6 (n)
= 4 f (2, λ2 , )
g( p, λ p )
p|n
= 4 f (2, λ2 , )
1−
p|n
g( p, λ p − 2)
g( p, λ p )
[g( p, λ p ) − g( p, λ p − 2)]
(2.24)
p
= 4 f (2, λ2 , )
p 2λ p +2 − (−1)(λ p +1)( p−1)/2
p 2 − (−1)( p−1)/2
p
p 2λ p −2 − (−1)(λ p −1)( p−1)/2
−
p 2 − (−1)( p−1)/2
p 2 − p −2
p 2 − (−1)( p−1)/2
p
(−1)( p−1)/2
= 4 f (2, λ2 , )
.
p 2λ p 1 +
p2
p
= 4 f (2, λ2 , )
p 2λ p
(2.25)
Now
f (2, λ2 , ) = 22λ2 +2 −
(−1)λ p ( p−1)/2
p
⎧
4−1=3
⎪
⎪
⎪
⎨4 + 1 = 5
=
⎪
16 − 1 = 15
⎪
⎪
⎩
16 + 1 = 17
if n ≡ 1
if n ≡ 3
if n ≡ 2
(mod 4)
(mod 4)
(mod 8)
if n ≡ 6
(mod 8).
Substituting this into (2.25) gives
p
r6 (n)
= c6 (n)n
2
p
(−1)( p−1)/2
1+
p2
where
⎧
12
⎪
⎪
⎪
⎨
20
c6 (n) =
⎪
15
⎪
⎪
⎩
17
Springer
if n ≡ 1
if n ≡ 3
(mod 4)
(mod 4)
if n ≡ 2
if n ≡ 6
(mod 8)
(mod 8).
,
On the number of primitive representations of integers as sums of squares
19
This completes the proof of Case 1.
Case 2. n ≡ 0 (mod 4)
The details are similar, but the prime 2 now plays a role in the Möbius formula
(1.3). In place of (2.24) we have
p
r6 (n) = 4( f (2, λ2 , ) − f (2, λ2 − 2, ))
(g( p, λ p ) − g( p, λ p − 2)). (2.26)
p
Observe that
15
× 22λ2 .
4
(2.27)
(−1)( p−1)/2
2λ p
1+
.
(g( p, λ p ) − g( p, λ p − 2)) =
p
p2
p
p
(2.28)
f (2, λ2 , ) − f (2, λ2 − 2, ) = 22λ2 +2 − 22λ2 −2 =
From the proof of Case 1, we also have
Substitute (2.27) and (2.28) into (2.26) to obtain
p
r6 (n)
(−1)( p−1)/2
15
2λ2
2λ p
1+
= 4×
p
×2
4
p2
p
(−1)( p−1)/2
2
= 15n
1+
.
p2
p
This proves Case 2 and completes the proof of Theorem 1 in the case k = 6.
3 Proof of the formulas for sums of an odd number of squares
We begin with
Lemma 4. Let the prime factorization of n be given by (1.5). Then
r3 (n) = r3 (m)
p λ p /2+1 − 1
p
p−1
−
−m
p
p λ p /2 − 1
,
p−1
(3.1)
Springer
20
S. Cooper, M. Hirschhorn
3λ2 /2+3
−1
2
23λ2 /2 − 1
r5 (n) = r5 (m)
(m)
+
5
23 − 1
23 − 1
3λ p /2
p 3λ p /2+3 − 1
−1
m p
×
−
p
,
p3 − 1
p
p3 − 1
p
5λ2 /2+5
−1
2
25λ2 /2 − 1
r7 (n) = r7 (m)
(m)
+
7
25 − 1
25 − 1
5λ p /2
p 5λ p /2+5 − 1
−1
p
2 −m
×
−
p
,
p5 − 1
p
p5 − 1
p
where
⎧
⎪
⎨
if m ≡ 1
0
(3.3)
(mod 8)
−4 if m ≡ 2 or 3
(mod 4)
⎪
⎩
−16/7 if m ≡ 5
(mod 8),
⎧
if m ≡ 1 or 2 (mod 4)
⎪
⎨ 8
0
if m ≡ 3
(mod 8)
7 (m) =
⎪
⎩
−64/37 if m ≡ 7
(mod 8).
5 (m) =
(3.2)
(3.4)
(3.5)
Also,
27λ2 /2+7 − 1 r9 (n) = r9 (m)
27 − 1
p
p 7λ p /2+7 − 1
− p3
p7 − 1
m
p
p 7λ p /2 − 1
,
p7 − 1
if n ≡ 5 (mod 8),
(3.6)
if n ≡ 7 (mod 8).
(3.7)
− 1 p 9λ p /2+9 − 1
2
−m p 9λ p /2 − 1
4
r11 (n) = r11 (m)
−
p
,
29 − 1
p9 − 1
p
p9 − 1
p
9λ2 /2+9
Proof: Equation (3.1) was proved in [22], Eqs. (3.2)–(3.6) were proved in [4] and
(3.7) was proved in [5]. All of the proofs used the theory of modular forms of half
integer weight.
Remark . Equation (3.1) was given explicitly by Pall [39]. Various special cases of
(3.1)–(3.7) have been given by many authors. For example, (this is by no means a
complete list) see Cohen [2], Hurwitz [23, 24], Mordell [35], Olds [36–38], Sandham
[43] and Stieltjes [50, 51]. For some additional results on sums of an odd number of
squares, see Bateman [1], Hardy [19, 20], Lomadze [30, 31] and Sandham [44].
Proof of Theorem 2 in the case k = 7
All of the formulas in Theorem 2 follow from (3.1)–(3.7) on using (1.3). We give
p
complete details for r7 (n); the other formulas have similar proofs.
Springer
On the number of primitive representations of integers as sums of squares
21
For λ ≥ 1, let
⎧ 5λ/2+5
25λ/2 − 1
−1
2
⎪
⎪
+ 7 (m) 5
⎪
⎨ 25 − 1
2 −1
h( p, λ, m) =
⎪
5λ/2+5
⎪
−1
−m p 5λ/2 − 1
p
⎪
2
⎩
−
p
p5 − 1
p
p5 − 1
if p = 2,
if p is an odd prime.
Observe that h( p, 1, m) = 1, and therefore Eq. (3.3) is equivalent to
25λ2 /2+5 − 1
25λ2 /2 − 1
+ 7 (m)
r7 (n) = r7 (m)
25 − 1
25 − 1
p 5λ p /2+5 − 1
−m p 5λ p /2 − 1
2
×
−
p
p5 − 1
p
p5 − 1
λ p ≥2
= r7 (m)h(2, λ2 , m)
h( p, λ p , m).
(3.8)
λ p ≥2
We consider two cases.
Case 1. n ≡ 0 (mod 4).
By Eqs. (1.3), (1.4) and (3.8), we have
p
r7 (n)
= r7 (n) −
r7
p12 |n
n
p12
+
r7
p12 , p22 |n
n
2 2
p1 p2
− · · · + (−1)
j
r7
p12 ,... , p 2j |n
n
2
p1 . . . p 2j
r7 n/ p12
r7 (n/ p 2 p 2 )
r7 (n/ p12 . . . p 2j )
1 2
j
= r7 (n) 1 −
+
− · · · + (−1)
r7 (n)
r7 (n)
r7 (n)
2
2 2
2
2
= r7 (m)
⎡
p1 |n
p1 , p2 |n
p1 ,... , p j |n
h( p, λ p , m)
λ p ≥2
× ⎣1 −
h( p1 , λ p − 2, m)
h( p1 , λ p − 2, m) h( p2 , λ p − 2, m)
1
1
2
+
h(
p
,
λ
,
m)
h(
p
,
λ
,
m)
h(
p
,
λ
1
p1
1
p1
2
p2 , m)
2
2 2
p1 |n
− · · · + (−1) j
p1 , p2 |n
p12 ,··· , p 2j |n
⎤
,
λ
−
2,
m)
h(
p
h( p1 , λ p1 − 2, m)
j
pj
⎦
···
h( p1 , λ p1 , m)
h( p j , λ p j , m)
Springer
22
S. Cooper, M. Hirschhorn
p
= r7 (m)
p
= r7 (m)
=
p
r7 (m)
h( p, λ p , m)
λ p ≥2
1−
λ p ≥2
h( p, λ p − 2, m)
h( p, λ p , m)
[h( p, λ p , m) − h( p, λ p − 2, m)]
(3.9)
λ p ≥2
5λ p /2
p 5λ p /2+5 − 1
−1
p
2 −m
−p
5
5
p −1
p
p −1
λ p ≥2
5λ p /2−5
p 5λ p /2 − 1
−1
p
2 −m
+
p
p5 − 1
p
p5 − 1
p
5λ p /2
2 −m
5λ p /2−5
= r7 (m)
p
−p
p
p
λ p ≥2
−m p
p
5λ p /2
= r7 (m)
1−
p
3
p
λ p ≥2
λ p ≥2
−m n 5/2 p
p
= r7 (m) 5/2
1−
.
3
m
p
n
p|
−
(3.10)
m
This completes the proof of Case 1.
Case 2. n ≡ 0 (mod 4).
We proceed as for Case 1, but this time the prime 2 plays a role in the Möbius
formula (1.3). In place of (3.9), we have
p
p
r7 (n) = r7 (m)[h(2, λ2 , m) − h(2, λ2 − 2, m)]
[h( p, λ p , m) − h( p, λ p − 2, m)].
λ p ≥2
(3.11)
Now using (3.5), we have
h(2, λ2 , m) − h(2, λ2 − 2, m)
=
25λ2 /2+5 − 1
25λ2 /2 − 1 25λ2 /2 − 1
25λ2 /2−5 − 1
(m)
(m)
+
−
−
7
7
25 − 1
25 − 1
25 − 1
25 − 1
= 25λ2 /2 + 7 (m)25λ2 /2−5
7 (m)
5λ2 /2
1+
=2
32
⎧
5
⎪
5λ /2
⎪
if m ≡ 1 or 2 (mod 4),
⎪2 2 ×
⎪
⎨
4
if m ≡ 3
(mod 8),
= 25λ2 /2 × 1
⎪
⎪
⎪
35
⎪
⎩25λ2 /2 ×
if m ≡ 7
(mod 8).
37
Springer
On the number of primitive representations of integers as sums of squares
23
Now substitute this into (3.11), and continue as in the proof of Case 1. This proves
Case 2, and completes the proof of Theorem 2 in the case k = 7.
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Springer