Pacific
Journal of
Mathematics
THE CONVOLUTION SUM
P
m<n/8 σ (m)σ (n−8m)
K ENNETH S. W ILLIAMS
Volume 228
No. 2
December 2006
PACIFIC JOURNAL OF MATHEMATICS
Vol. 228, No. 2, 2006
THE CONVOLUTION SUM
P
m<n/8 σ (m)σ (n−8m)
K ENNETH S. W ILLIAMS
P
The convolution sum m<n/8 σ (m)σ (n−8m) is evaluated for all n ∈ N. This
evaluation is used to determine the number of representations of n by the
quadratic form x12 + x22 + x32 + x42 + 2x52 + 2x62 + 2x72 + 2x82 .
1. Introduction
Let N denote the set of natural numbers. For n ∈ N and k ∈ N, we set
X
σk (n) =
dk,
d|n
where d runs through the positive integers dividing n. If n ∈
/ N, we set σk (n) = 0.
We write σ (n) for σ1 (n). We define the convolution sum Wk (n) by
X
Wk (n) :=
(1)
σ (m)σ (n−km),
m<n/k
where m runs through the positive integers < n/k. The sum Wk (n) has been evaluated for k = 1, 2, 3, 4, 9 for all n ∈ N, and for k = 5 for n ≡ 8 (mod 16), n 6≡ 0
(mod 5); see [Besge 1862; Huard et al. 2002] for k = 1, [Huard et al. 2002; Melfi
1998a; 1998b] for k = 2, [Huard et al. 2002; Melfi 1998a; 1998b; Williams 2004]
for k = 3, [Huard et al. 2002; Melfi 1998a; 1998b] for k = 4, [Melfi 1998a; 1998b;
Williams 2004; 2005] for k = 9, and [Melfi 1998a; 1998b] for k = 5. In this paper,
we evaluate Wk (n) for k = 8 and all n ∈ N.
Let Z, R, C denote the sets of integers, real numbers, and complex numbers,
respectively. For q ∈ C with |q| < 1, we set [Ramanujan 1916, eqn. (92), p. 151]
(2)
1(q) := q
∞
Y
(1 − q n )24 .
n=1
MSC2000: 11A25, 11E20, 11E25.
Keywords: divisor functions, convolution sums, Eisenstein series.
Research supported by Natural Sciences and Engineering Research Council of Canada grant A-7233.
387
388
KENNETH S. WILLIAMS
For n ∈ N, we define k(n) ∈ Z by
1(q )1(q )
2
(3)
4
1/6
=q
∞
Y
(1 − q ) (1 − q ) :=
2n 4
4n 4
n=1
∞
X
k(n)q n .
n=1
Clearly, k(n) = 0 for n ≡ 0 (mod 2). By Euler’s identity [Knopp 1970, Corollary
5, p. 37], we have
∞
Y
(1 − q n ) =
∞
X
(−1)m q m(3m+1)/2 = 1 − q − q 2 + q 5 + q 7 − q 12 − · · · .
m=−∞
n=1
Using MAPLE, we find
q
∞
Y
(1 − q 2n )4 (1 − q 4n )4 = q − 4q 3 − 2q 5 + 24q 7 − 11q 9 − · · · ,
n=1
so that the first few values of k(2n−1), for n ∈ N, are
k(1) = 1,
k(3) = −4,
k(7) = 24,
k(5) = −2,
k(9) = −11.
The evaluation of W8 (n) is given in Theorem 1 and proved in Section 2.
Theorem 1. For n ∈ N,
X
σ (m)σ (n−8m) =
m<n/8
1
1 n 1 n 1 n σ3 (n) + σ3
+ σ3
+ σ3
192
64
2
16
4
3
8
1
1
n
n n 1
+
−
σ (n) +
−
σ
− k(n).
24 32
24 4
8
64
As an application of Theorem 1, we evaluate in Section 3 the number
N (n) = card (x1 , x2 , . . . , x8 ) ∈ Z8 n = x12 +x22 +x32 +x42 +2x52 +2x62 +2x72 +2x82 .
We prove:
Theorem 2. For n ∈ N,
N (n) = 4σ3 (n) − 4σ3
n 2
− 16σ3
n 4
+ 256σ3
n 8
+ 4k(n).
THE CONVOLUTION SUM
m<n/8 σ (m)σ (n−8m)
P
389
2. Proof of Theorem 1
Let q ∈ C be such that |q| < 1. The Eisenstein series L(q), M(q), N (q) are
L (q) := 1 − 24
(4)
∞
X
σ (n)q n ,
n=1
∞
X
(5)
M(q) := 1 + 240
N (q) := 1 − 504
(6)
n=1
∞
X
σ3 (n)q n ,
σ5 (n)q n ,
n=1
see for example [Berndt 1989, p. 318; Ramanujan 1916, eqn. (25), p. 140]. It was
shown in [Ramanujan 1916, eqn. (44)] that
1
M(q)3 − N (q)2 .
(7)
1(q) =
1728
First, we determine the generating function of W8 (n) for n ∈ N. By (4), we have
1 − L(q) 1 − L(q 8 )
∞
∞
∞
X
X
X
l
8m
= 24
σ (l)q 24
σ (m)q
= 576
σ (l)σ (m)q l+8m
= 576
l=1
∞
X
n=1
that is,
m=1
qn
∞
X
σ (l)σ (m) = 576
l,m=1
l+8m=n
qn
X
σ (m)σ (n−8m),
n=1 1≤m<n/8
∞
X
8
W8 (n)q n ,
1 − L(q) 1 − L(q ) = 576
n=1
so that
(8)
l,m=1
∞
X
∞
X
n=1
W8 (n)q n =
1
1
1
1
−
L(q) −
L(q 8 ) +
L(q) L(q 8 ).
576 576
576
576
Next, we use (4) and (8) to determine the power-series expansion of L(q) L(q 8 ) in
powers of q. Letting q 7→ q 8 in (4) yields
(9)
L(q 8 ) = 1 − 24
∞
X
n=1
σ (n)q 8n = 1 − 24
∞
n X
σ
qn.
8
n=1
From (4), (8), and (9) we deduce
∞ n X
(10)
L(q) L(q 8 ) = 1 +
576 W8 (n) − 24σ (n) − 24σ
qn.
8
n=1
390
KENNETH S. WILLIAMS
2
Our next objective is to determine the power series of L(q) − 8 L(q 8 ) in
powers of q. The result,
(11)
L(q)2 = 1 +
∞
X
240σ3 (n) − 288n σ (n) q n ,
n=1
is classical, see for example [Glaisher 1885a; 1885b]. Letting q 7→ q 8 in (11), we
obtain
∞ n n X
8 2
(12)
L(q ) = 1 +
240σ3
− 36n σ
qn.
8
8
n=1
Hence, by (10), (11), and (12), we obtain
2
L(q) − 8 L(q 8 ) = L(q)2 + 64 L(q 8 )2 − 16 L(q) L(q 8 )
= 1+
∞
X
(240σ3 (n) − 288n σ (n))q n
n=1
+ 64 1 +
− 16 1 +
∞ X
n=1
∞ X
240σ3
n 8
− 36n σ
n 8
q
576 W8 (n) − 24σ (n) − 24σ
n=1
n
n 8
q
n
,
that is,
(13)
2
L(q) − 8 L(q 8 )
∞ n X
= 49 +
240σ3 (n) + 15360σ3
+ (384 − 288n)σ (n)
8
n=1
n + (384 − 2304n)σ
− 9216 W8 (n) q n .
8
Next, we use Ramanujan’s evaluation of L(q) [Berndt 1991, p. 129] and the
principle of duplication [Berndt 1991, p. 125] to determine the power-series ex2
pansion of L(q) − 8 L(q 8 ) in another way. For z ∈ C with |z| < 1, we set
1
1
∞
X
n
1 1
2 n 2 n z ,
(14)
w(z) = 2 F1 , ; 1; z =
2 2
(1)n
n!
n=0
where 2 F1 is the Gaussian hypergeometric function and (a)n is the Pochhammer
symbol, see for example [Copson 1935, p. 247; Rainville 1971, p. 45]. Clearly,
w(0) = 1. The infinite series (14) diverges at z = 1 [Copson 1935, p. 249], so that
THE CONVOLUTION SUM
m<n/8 σ (m)σ (n−8m)
P
391
w(1) = +∞. For x ∈ R with 0 ≤ x < 1, we have
∞
X
(2n!)2 n
w(x) = 1 +
x ≥1
(n!)4 24n
n=1
so that
w(x) 6= 0,
(15)
0 ≤ x < 1.
The derivative with respect to x of the function
y(x) := π
(16)
w(1 − x)
,
w(x)
0 < x < 1,
is
y 0 (x) =
(17)
−1
,
x(1 − x)w(x)2
0 < x < 1,
see [Berndt 1989, p. 87]. Thus, by (15) and (17), we have
y 0 (x) < 0,
(18)
0 < x < 1.
Hence, as x increases from 0 to 1, y(x) strictly decreases
from
y(0) = π
w(1)
= +∞
w(0)
to
y(1) = π
w(0)
= 0.
w(1)
Now, restrict q so that q ∈ R and 0 < q < 1. Thus, 0 < − log q < +∞. Hence,
there is a unique value of x between 0 and 1 such that
(19)
y(x) = − log q.
Ramanujan gave in his notebooks [Ramanujan 1957] the following formulae for
L(q), M(q) and N (q), which are proved in [Berndt 1991, pp. 126–129]:
(20)
L (q) = (1 − 5x)w2 + 12x(1 − x)w
(21)
M(q) = (1 + 14x + x 2 )w4 ,
(22)
N (q) = (1 + x)(1 − 34x + x 2 )w 6 .
dw
,
dx
From (7), (21), and (22), we obtain
(23)
1(q) =
x(1 − x)4 w 12
.
24
Applying the principle of duplication [Berndt 1991, p. 125]
√
√
1− 1−x 2
1+ 1−x
2
q 7→ q ,
x 7→
,
w
7
→
w
√
2
1+ 1−x
392
KENNETH S. WILLIAMS
to (20), (21), and (23), we obtain
(24)
L (q 2 ) = (1 − 2x)w 2 + 6x(1 − x)w
(25)
M(q 2 ) = (1 − x + x 2 )w 4 ,
dw
,
dx
x 2 (1 − x)2 w 12
.
28
Formulae (24) and (25) are given in [Berndt 1991, pp. 122, 126]. Applying the
principle of duplication to (24), (25), and (26), we obtain
5 dw
L (q 4 ) = 1 − x w 2 + 3x(1 − x)w
(27)
,
4
dx
1
(28)
M(q 4 ) = 1 − x + x 2 w 4 ,
16
x 4 (1 − x)w 12
1(q 4 ) =
(29)
.
216
Formulae (27) and (28) can be deduced from [Berndt 1991, pp. 122, 127]. From
(3), (26), and (29), we obtain
(26)
(30)
1(q 2 ) =
∞
X
√
x 1 − x w4 = 16
k(n)q n .
n=1
Applying the duplication principle to (27) and (28), we have
5 11
3√
3
dw
(31)
− x+
1 − x w2 + x(1 − x)w
,
L (q 8 ) =
8 16
8
2
dx
17 17
1 2 15 √
15 √
− x+
x +
(32)
M(q 8 ) =
1 − x − x 1 − x w4 ,
32 32
256
32
64
see [Cheng and Williams 2004, p. 564]. From (20) and (31), we deduce
√
1
(33)
L(q) − 8 L(q 8 ) = −4 + x − 3 1 − x w2 .
2
Squaring both sides of (33), we have
√
√
2 1
(34)
L(q) − 8 L(q 8 ) = 25 − 13x + x 2 + 24 1 − x − 3x 1 − x w4 .
4
From (21), (25), and (28), we obtain
(35)
(36)
(37)
1
2
16
M(q) − M(q 2 ) + M(q 4 ),
15
15
15
1
1
xw 4 =
M(q) − M(q 2 ),
15
15
16
16
x 2 w4 =
M(q 2 ) − M(q 4 ).
15
15
w4 =
THE CONVOLUTION SUM
m<n/8 σ (m)σ (n−8m)
P
393
Using (30), (35), (36), and (37) in (32), we obtain
∞
M(q 8 ) = −
15 X
1
9
15 √
1 − x w4 −
M(q 2 ) + M(q 4 ) +
k(n)q n .
32
16
32
4
n=1
Thus,
√
(38)
∞
1 − x w4 =
X
6
32
1
M(q 2 ) − M(q 4 ) + M(q 8 ) + 8
k(n)q n .
15
5
15
n=1
Now, using (30), (35), (36), (37), and (38) in (34), we deduce
(39)
2
L(q) − 8 L(q 8 )
∞
=
X
4
12
256
3
k(n)q n .
M(q) − M(q 2 ) − M(q 4 ) +
M(q 8 ) + 144
5
5
5
5
n=1
Appealing to (5) and (39), we obtain
(40)
∞ n n X
2
− 576σ3
L(q) − 8 L(q 8 ) = 49 +
192σ3 (n) − 144σ3
2
4
n=1
n + 12288σ3
+ 144k(n) q n .
8
Equating coefficients of q n in (13) and (40), we deduce that
n +(384−288n)σ (n)+(384−2304n)σ
−9216 W8 (n)
8
8 n n n
= 192σ3 (n) − 144σ3
− 576σ3
+ 12288σ3
+ 144k(n),
2
4
8
240σ3 (n)+15360σ3
n from which the asserted formula for W8 (n) follows.
3. The number of representations of n by the quadratic form
x12 + x22 + x32 + x42 + 2x52 + 2x62 + 2x72 + 2x82
Proof of Theorem 2. Let N0 = N ∪ {0}. For l ∈ N0 , we set
r4 (l) = card (x1 , x2 , x3 , x4 ) ∈ Z4 x12 + x22 + x32 + x42 = l ,
so that r4 (0) = 1. The number N (n) of representations of n by the quadratic form
x12 + x22 + x32 + x42 + 2x52 + 2x62 + 2x72 + 2x82 is
394
KENNETH S. WILLIAMS
N (n) = card (x1 , . . . , x8 ) ∈ Z8 x12 + x22 + x32 + x42 + 2x52 + 2x62 + 2x72 + 2x82 = n
X
X X
X
=
1
1 =
r4 (l)r4 (m)
l,m∈N0 (x1 ,x2 ,x3 ,x4 )∈Z4
l+2m=n x 2 +x 2 +x 2 +x 2 =l
1
2
3
4
= r4 (0)r4
n 2
= r4 (n) + r4
(x5 ,x6 ,x7 ,x8 )∈Z4
x52 +x62 +x72 +x82 =m
X
+ r4 (n)r4 (0) +
n 2
l,m∈N0
l+2m=n
r4 (l)r4 (m)
l,m∈N
l+2m=n
+
X
r4 (l)r4 (m).
l,m∈N
l+2m=n
It is a classical result of Jacobi — see for example [Spearman and Williams 2000] —
that
n X
r4 (n) = 8
,
n ∈ N.
d = 8σ (n) − 32σ
4
d|n
4-d
Hence,
N (n) = 8σ (n) − 32σ
n 4
n − 32σ
2
8
l m X
+
8σ (l) − 32σ
8σ (m) − 32σ
.
4
4
+ 8σ
n l,m∈N
l+2m=n
Thus,
N (n) − 8σ (n) − 8σ
n = 64
+ 32σ
2
X
n 4
+ 32σ
n 8
X
σ (l)σ (m) − 256
l,m∈N
l+2m=n
σ
l,m∈N
l+2m=n
−256
X
σ (l)σ
= 64
4
m 4
l,m∈N
l+2m=n
X
l σ (m)σ (n − 2m) − 256
σ (m)
+ 1024
σ
l,m∈N
l+2m=n
X
l m σ
4
4
σ (l)σ (m)
l,m∈N
4l+2m=n
m<n/2
−256
X
σ (l)σ (m) + 1024
l,m∈N
l+8m=n
= 64 W2 (n) − 256
X
l<n/4
−256
X
X
m<n/8
σ (l)σ
n
2
X
σ (l)σ (m)
l,m∈N
4l+8m=n
− 2l
σ (m)σ (n−8m) + 1024
X
m<n/8
σ (m)σ
n
4
− 2m ,
THE CONVOLUTION SUM
m<n/8 σ (m)σ (n−8m)
P
395
that is,
n n − 32σ
2
4
n 8
n + 64 W2 (n) − 256 W2
− 256 W8 (n) + 1024 W2
.
2
4
From [Huard et al. 2002, Theorem 2], we have
1
1 n 1
n
n n 1
(42)
+
−
σ (n) +
−
σ
.
W2 (n) = σ3 (n) + σ3
12
3
2
24 8
24 4
2
Appealing to (41), (42), and Theorem 1, we obtain
n n n N (n) = 4σ3 (n) − 4σ3
− 16σ3
+ 256σ3
+ 4k(n),
2
4
8
as claimed.
(41) N (n) = 8σ (n) + 8σ
− 32σ
n The values of N (n), σ3 (n) and k(n) for n = 1, 2, . . . , 10 are as follows:
n
N (n)
σ3 (n)
k(n)
1
8
1
1
2
32
9
0
3
96
28
−4
4
240
73
0
5
496
126
−2
6
896
252
0
7
1472
344
24
8
2160
585
0
9
2984
757
−11
10
4032
1134
0
Note added in proof
Since this paper was written, the sums Wk (n) have been evaluated for k = 5, 6, 7,
12, 16, 18 and 24. See M. Lemire and K. S. Williams, Bull. Austral. Math. Soc.
73 (2006), 107–115; A. Alaca, S. Alaca and K. S. Williams, Adv. Th. App. Math.
1 (2006), 27–48, Int. Math. Forum 2 (2007), 45–68, Math. J. Okayama Univ. (in
press), Canad. Math. Bull. (to appear); and S. Alaca and K. S. Williams, J. Number
Th. (in press).
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Received May 19, 2005. Revised July 13, 2005.
K ENNETH S. W ILLIAMS
C ENTRE FOR R ESEARCH IN A LGEBRA AND N UMBER T HEORY
S CHOOL OF M ATHEMATICS AND S TATISTICS
C ARLETON U NIVERSITY
OTTAWA , O NTARIO K1S 5B6
C ANADA
[email protected]
http://www.mathstat.carleton.ca/~williams