EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 3, 2014, 304-311
ISSN 1307-5543 – www.ejpam.com

Representation Number Formulae for Some Mixed Quadratic
Forms

Bülent Köklüce

Faculty of Education, Fatih University, Istanbul, Turkey

Abstract. We find formulae for the number of representations of a positive integer by some quadratic
forms which are sums of squares and the form x2

1 + x1 x2 + x2
2 .

2010 Mathematics Subject Classifications: 11A25,11E25
Key Words and Phrases: Quadratic Forms, Representation Numbers

1. Introduction

Let N , N0, Z and C denote the set of natural numbers, non-negative integers, integers and
complex numbers so that N0 = N∪ {0}. For k, n ∈ N we define

σk(n) :=
∑

d∈N
d|n

dk (1)

where d runs through the positive integers dividing n. We write σ(n) for σ1(n) =
∑

d∈N
d|n

d. If

n /∈ N, we set σk(n) = 0. For a1, . . . , a6 ∈ N and n ∈ N0 we define

M(a1, a2, a3, a4, a5, a6; n) := card







�

x1, . . . , x8

�

∈ Z8 :
n= a1 x2

1 + a2 x2
2 + a3 x2

3 + a4 x2
4

+a5(x2
5 + x5 x6 + x2

6) + a6(x2
7 + x7 x8 + x2

8)







. (2)

With our notation, the representation number formulae for

M(1,1, 1,1, 2,2; n), M(3, 3,3,3, 2,2; n), M(1,1, 1,1, 1,1; n), M(3, 3,3,3, 1,1; n),
M(1,1, 3,3, 1,1; n), M(1, 1,3,3, 2,2; n), M(1,1, 3,3, 4,4; n), M(1, 1,3,3, 1,2; n),
M(1,1, 3,3, 1,4; n), M(1, 1,3,3, 2,4; n), M(1,1, 1,1, 1,2; n), M(3, 3,3,3, 1,2; n),
M(1,1, 1,1, 1,4; n), M(3, 3,3,3, 1,4; n), M(1,1, 1,1, 2,4; n), M(3,3, 3,3, 2,4; n)

Email address: bkokluce@fatih.edu.tr

http://www.ejpam.com 304 c© 2014 EJPAM All rights reserved.



B. Köklüce / Eur. J. Pure Appl. Math, 7 (2014), 304-311 305

are given in [5] respectively by the equations from (2.10) to (2.25). In that study the authors
firstly uses the (p − k) parametrization of theta functions given by Alaca, Alaca and Williams
to establish some new theta function identities and then uses them to determine formulae for
the number of representations of positive integers by certain quadratic forms.

The aim of this paper is to determine explicit formulae for

M(1, 1,2, 2,2, 2; n), M(2,2, 2,2, 1,1; n), M(3, 3,6, 6,4, 4; n), M(1,1, 1,1, 6,6; n),
M(3, 3,6, 6,2, 2; n), M(1,1, 2,2, 1,1; n), M(1, 1,2, 2,4, 4; n), M(3,3, 6,6, 1,1; n),
M(1, 1,3, 3,6, 6; n), M(1,2, 2,4, 2,2; n), M(1, 2,2, 4,4,4; n)

.

We use some known representation number formulae for quaternary quadratic forms and
convolutions sums of divisors. The method firstly have been used by Alaca, Alaca and Williams.
The quadratic forms considered here have not been considered before. As in [5] the formulae
are given in terms of σ3(n) and the numbers c1,6(n), c1,8(n), c1,12(n), c3,4(n), c1,18(n), c2,9(n),
c1,24(n), c3,8(n).

The representation numbers for the quaternary quadratic forms

f1 :=x2
1 + x2

2 + x2
3 + x2

4 , (3)

f2 :=x2
1 + x2

2 + 2x2
3 + 2x2

4 , (4)

f3 :=x2
1 + x2

2 + 3x2
3 + 3x2

4 , (5)

f4 :=x2
1 + 2x2

2 + 2x2
3 + 4x2

4 , (6)

f5 :=x2
1 + x1 x2 + x2

2 + x2
3 + x3 x4 + x2

4 , (7)

have been determined before. For l ∈ N0, if we set

ri(l) = card
��

x1, . . . , x4

�

∈ Z4 : l = fi(x1, x2, x3, x4)
	

then clearly ri(0) = 1 for each i ∈ {1,2, 3,4, 5}. It is known (see for example [4] for the first
four and [7] for the last one) that for l ∈ N,

r1(l) =8σ(l)− 32σ(
l
4
), (8)

r2(l) =4σ(l)− 4σ(
l
2
) + 8σ(

l
4
)− 32σ(

l
8
), (9)

r3(l) =4σ(l)− 8σ(
l
2
)− 12σ(

l
3
) + 16σ(

l
4
) + 24σ(

l
6
)− 48σ(

l
12
), (10)

r4(l) =2σ(l)− 2σ(
l
2
) + 8σ(

l
8
)− 32σ(

l
16
), (11)

r5(l) =12σ(l)− 36σ(
l
3
). (12)

For r, s, n ∈ N with r ≤ s the convolution sum Wr,s(n) is defined by

Wr,s(n) :=
∑

(l,m)∈N2
0

r l+sm=n

σ(l)σ(m).



B. Köklüce / Eur. J. Pure Appl. Math, 7 (2014), 304-311 306

The sum Wr,s(n) has been evaluated for certain values, see [6, 7, 10] for W1,1(n), [7] for
W1,2(n) and W1,4(n), [7–9, 11] for W1,3(n), [3] for W1,6(n) and W2,3(n), [12] for W1,8(n), [1]
for W1,12(n) and W3,4(n), and [2] for W1,24(n) and W3,8(n).

Theorem 1. Let n ∈ N then,

(i)

M(1,1, 2,2, 2,2; n) =
14
5
σ3(n)−

14
5
σ3(

n
2
)−

54
5
σ3(

n
3
) +

28
5
σ3(

n
4
) +

54
5
σ3(

n
6
)

−
448

5
σ3(

n
8
)−

108
5
σ3(

n
12
) +

1728
5
σ3(

n
24
) +

6
5

c1,6(n)

+
12
5

c1,6(
n
2
)−

12
5

c3,4(
n
2
),

(ii)

M(2, 2,2,2, 1,1; n) =
21
5
σ3(n) +

91
5
σ3(

n
2
)−

81
5
σ3(

n
3
)−

84
5
σ3(

n
4
)−

351
5
σ3(

n
6
)

−
448
5
σ3(

n
8
) +

324
5
σ3(

n
12
) +

1728
5
σ3(

n
24
) +

12
5

c1,6(n)

+ 6c1,8(n)−
3
5

c3,8(n),

(iii)

M(3,3, 6,6, 4,4; n) =
1
10
σ3(n)−

1
10
σ3(

n
2
)−

21
10
σ3(

n
3
) +

4
5
σ3(

n
4
) +

21
10
σ3(

n
6
)

−
64
5
σ3(

n
8
)−

84
5
σ3(

n
12
) +

1344
5
σ3(

n
24
) +

2
5

c1,6(
n
2
)

+
16
5

c1,6(
n
4
)−

1
10

c3,4(n),

(iv)

M(1, 1,1, 1,6, 6; n) =
8
15
σ3(n) +

76
15
σ3(

n
3
)−

128
15
σ3(

n
4
)−

108
5
σ3(

n
9
)

−
1216

15
σ3(

n
12
) +

1728
5
σ3(

n
36
)−

4
5

c1,6(n) +
16
5

c1,6(
n
2
)

+
124

5
c1,18(n)−

16
15

c2,9(
n
2
),

(v)

M(3,3, 6,6, 2,2; n) =
2
5
σ3(n)−

2
5
σ3(

n
2
)−

42
5
σ3(

n
3
) +

4
5
σ3(

n
4
) +

42
5
σ3(

n
6
)

−
64
5
σ3(

n
8
)−

84
5
σ3(

n
12
) +

1344
5
σ3(

n
24
)−

2
5

c1,6(n)

−
4
5

c1,6(
n
2
) +

44
5

c1,12(
n
2
),



B. Köklüce / Eur. J. Pure Appl. Math, 7 (2014), 304-311 307

(vi)

M(1,1, 2,2, 1,1; n) =
56
5
σ3(n)−

56
5
σ3(

n
2
)−

216
5
σ3(

n
3
) +

28
5
σ3(

n
4
) +

216
5
σ3(

n
6
)

−
448
5
σ3(

n
8
)−

108
5
σ3(

n
12
) +

1728
5
σ3(

n
24
)−

6
5

c1,6(n)

+ 6c1,8(n) +
3
5

c3,4(n)−
3
5

c3,8(n),

(vii)

M(1, 1,2, 2,4, 4; n) =
7
10
σ3(n)−

7
10
σ3(

n
2
)−

27
10
σ3(

n
3
) +

28
5
σ3(

n
4
) +

27
10
σ3(

n
6
)

−
448

5
σ3(

n
8
)−

108
5
σ3(

n
12
) +

1728
5
σ3(

n
24
)−

6
5

c1,6(
n
2
)

−
48
5

c1,6(
n
4
) +

33
10

c1,12(n),

(viii)

M(3,3, 6,6, 1,1; n) =
8
5
σ3(n)−

8
5
σ3(

n
2
)−

168
5
σ3(

n
3
) +

4
5
σ3(

n
4
) +

168
5
σ3(

n
6
)

−
64
5
σ3(

n
8
)−

84
5
σ3(

n
12
) +

1344
5
σ3(

n
24
) +

2
5

c1,6(n)

− 18c1,8(
n
3
)−

11
5

c1,12(n) +
61
5

c1,24(n),

(ix)

M(1,1, 3,3, 6,6; n) =
4
15
σ3(n)−

8
15
σ3(

n
2
)−

88
15
σ3(

n
3
) +

64
15
σ3(

n
4
) +

176
15
σ3(

n
6
)

+
108
5
σ3(

n
9
)−

1408
15

σ3(
n

12
)−

216
5
σ3(

n
18
)−

2
5

c1,6(n)

−
8
5

c1,6(
n
2
)−

18
5

c1,6(
n
3
) +

1728
5
σ3(

n
36
)−

72
5

c1,6(
n
6
)

−
16
3

c1,9(
n
2
) +

62
15

c1,18(n) +
8

15
c2,9(

n
2
),

(x)

M(1, 2,2, 4,2, 2; n) =
7
5
σ3(n)−

7
5
σ3(

n
2
)−

27
5
σ3(

n
3
) +

27
5
σ3(

n
6
) +

28
5
σ3(

n
8
)

−
448

5
σ3(

n
16
)−

108
5
σ3(

n
24
) +

1728
5
σ3(

n
48
) +

3
5

c1,6(n)

+ 6c1,8(
n
2
) +

3
5

c3,4(
n
2
)−

3
5

c3,8(
n
2
),



B. Köklüce / Eur. J. Pure Appl. Math, 7 (2014), 304-311 308

(xi)

M(1,2, 2,4, 4,4; n) =
7

20
σ3(n)−

7
20
σ3(

n
2
)−

27
20
σ3(

n
3
) +

27
20
σ3(

n
6
) +

28
5
σ3(

n
8
)

−
448

5
σ3(

n
16
)−

108
5
σ3(

n
24
) +

1728
5
σ3(

n
48
)−

3
5

c1,6(
n
2
)

+
12
5

c1,6(
n
4
) +

33
20

c1,12(n)−
12
5

c3,4(
n
4
),

2. Proof of Theorem 1

In this section we give the proof of Theorem 1. (i). The remaining parts can be proved by
a similar way.

Proof. [(i)] The form f := x2
1+x2

2+2x2
3+2x2

4+2x2
5+2x5 x6+2x2

6+2x2
7+2x7 x8+2x2

8 clearly
can be obtained from the sum of the quaternary quadratic forms f2 := x2

1 + x2
2 + 2x2

3 + 2x2
4

and f5 := x2
1 + x1 x2 + x2

2 + x2
3 + x3 x4 + x2

4 . It is obvious that

M(1,1, 2,2, 2,2; n) =
∑

l,m∈N0
l+2m=n

r2(l)r5(m)

=r1(0)r5(
n
2
) + r1(n)r5(0) +

∑

l,m∈N
l+2m=n

r2(l)r5(m).

Thus by using the equations (9) and (12) we have

M(1,1, 2,2, 2,2; n)−(4σ(n)− 4σ(
n
2
) + 8σ(

n
4
)− 32σ(

n
8
) + 12σ(n)− 36σ(

n
3
))

=
∑

l,m∈N
l+2m=n

(4σ(l)− 4σ(
l
2
) + 8σ(

l
4
)− 32σ(

l
8
))(12σ(m)− 36σ(

m
3
))

=48
∑

l,m∈N
l+2m=n

σ(l)σ(m)− 144
∑

l,m∈N
l+2m=n

σ(l)σ(
m
3
)− 48

∑

l,m∈N
l+2m=n

σ(
l
2
)σ(m)

+ 144
∑

l,m∈N
l+2m=n

σ(
l
2
)σ(

m
3
) + 96

∑

l,m∈N
l+2m=n

σ(
l
4
)σ(m)− 288

∑

l,m∈N
l+2m=n

σ(
l
4
)σ(

m
3
)

− 384
∑

l,m∈N
l+2m=n

σ(
l
8
)σ(m) + 1156

∑

l,m∈N
l+2m=n

σ(
l
8
)σ(

m
3
).

So,

M(1, 1,2, 2,2, 2; n) =4σ(n)− 4σ(
n
2
) + 8σ(

n
4
)− 32σ(

n
8
) + 12σ(n)



B. Köklüce / Eur. J. Pure Appl. Math, 7 (2014), 304-311 309

− 36σ(
n
3
) + 48W1,2(n)− 144W1,6(n)− 48W1,1(

n
2
)

+ 144W1,3(
n
2
) + 96W1,2(

n
2
)− 288W2,3(

n
2
)

− 384W1,4(
n
2
) + 1156W3,4(

n
2
) (13)

where

W1,1(n) =
5

12
σ3(n)−

n
2
σ(n) +

1
12
σ(n), (14)

W1,2(n) =
1

12
σ3(n) +

1
3
σ3(

n
2
)−

n
8
σ(n)−

n
4
σ(

n
2
) +

1
24
σ(n) +

1
24
σ(

n
2
), (15)

W1,3(n) =
1

24
σ3(n) +

3
8
σ3(

n
3
)−

1
12

nσ(n)−
1
4

nσ(
n
3
) +

1
24
σ(n)

+
1
24
σ(

n
3
), (16)

W1,4(n) =
1

48
σ3(n) +

1
16
σ3(

n
2
) +

1
3
σ(

n
4
)−

n
16
σ(n)−

n
4
σ(

n
4
) +

1
24
σ(n)

+
1
24
σ(

n
4
), (17)

W1,6(n) =
1

120
σ3(n) +

1
30
σ3(

n
2
) +

3
40
σ3(

n
3
) +

3
10
σ3(

n
6
) + (

1
24
−

n
24
)σ(n)

+ (
1
24
−

n
4
)σ(

n
6
)−

1
120

c1,6(n) (18)

W2,3(n) =
1

120
σ3(n) +

1
30
σ3(

n
2
) +

3
40
σ3(

n
3
) +

3
10
σ3(

n
6
) + (

1
24
−

n
12
)σ(

n
2
)

+ (
1
24
−

n
8
)σ(

n
3
)−

1
120

c1,6(n), (19)

where
∞
∑

n=1

c1,6(n)q
n = q

∞
∏

n=1

(1− qn)2(1− q2n)2(1− q3n)2(1− q6n)2, (20)

and

W3,4(n) =
1

480
σ3(n) +

1
160

σ3(
n
2
) +

3
160

σ3(
n
3
) +

1
30
σ3(

n
4
) +

9
160

σ3(
n
6
)

+
3
10
σ3(

n
12
) + (

1
24
−

n
16
)σ(

n
3
) + (

1
24
−

n
12
)σ(

n
4
)

−
1

480
c3,4(n), (21)

where

∞
∑

n=1

c3,4(n)q
n =10q2

∞
∏

n=1

(1− qn)3(1− q2n)2(1− q3n)−1(1− q4n)−1(1− q6n)2(1− q12n)3



REFERENCES 310

+ q
∞
∏

n=1

(1− qn)−2(1− q2n)8(1− q3n)−2(1− q4n)−2(1− q6n)8(1− q12n)−2.

(22)

In any formula n ∈ N and q ∈ C. Substituting (14)-(22) in (13) gives

M(1, 1,2,2, 2,2; n) =
14
5
σ3(n)−

14
5
σ3(

n
2
)−

54
5
σ3(

n
3
) +

28
5
σ3(

n
4
) +

54
5
σ3(

n
6
)

−
448

5
σ3(

n
8
)−

108
5
σ3(

n
12
) +

1728
5
σ3(

n
24
) +

6
5

c1,6(n)

+
12
5

c1,6(
n
2
)−

12
5

c3,4(
n
2
), (23)

which is the asserted formula.

Denoting the right hand side of (23) by F(n), we give a list of values of M(1,1, 1,1, 2,2; n)
and F(n) in Table 1 to illustrate the equations.

Table 1: The first ten values of M(1, 1,1, 1,2, 2; n) and F(n)

n M(1, 1,2,2, 2,2; n) σ3(n) c1,6(n) c1,6(
n
2 ) c3,4(

n
2 ) F(n)

1 4 1 1 0 0 4
2 20 9 −2 1 1 20
3 64 28 −3 0 0 64
4 156 73 4 −2 12 156
5 360 126 6 0 0 360
6 620 252 6 −3 −33 620
7 944 344 −16 0 0 944
8 1452 585 −8 4 −24 1452
9 1828 757 9 0 0 1828

10 2520 1134 −12 6 126 2520

References

[1] A. Alaca, Ş. Alaca, and K. S. Williams. Evaluation of the convolution sums
∑

l+12m=n
σ(l)σ(m) and

∑

3l+4m=n

σ(l)σ(m), Advances in Theoretical and Applied Mathemat-

ics, 1, 1, 27–48. 2006.

[2] A. Alaca, Ş. Alaca, and K. S. Williams. Evaluation of the convolution sums
∑

l+24m=n
σ(l)σ(m) and

∑

3l+8m=n

σ(l)σ(m), Mathematical Journal of Okayama University,

49, 93–111. 2007.

[3] Ş. Alaca and K. S. Williams. Evaluation of the convolution sums
∑

l+6m=n
σ(l)σ(m) and

∑

2l+3m=n

σ(l)σ(m), Journal of Number Theory. 124, 491–510. 2007.



REFERENCES 311

[4] A. Alaca, Ş. Alaca, M. F. Lemire, and K. S. Williams. Nineteen quaternary quadratic forms,
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[5] X. W. X. Ernest and X. M. Y. Olivia. On the representations of integers by certain quadratic
forms, International Journal of Number Theory, 9, 189–204. 2013.

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