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Georgian Mathematical Journal
Volume 8 (2001), Number 1, 111-127

ON THE NUMBER OF REPRESENTATIONS OF POSITIVE
INTEGERS BY THE QUADRATIC FORM x21 + · · · + x28 + 4x29
G. LOMADZE

Abstract. An explicit exact (non asymptotic) formula is derived for the
number of representations of positive integers by the quadratic form x21 +
· · ·+x28 +4x29 . The way by which this formula is derived, gives us a possibility
to develop a method of finding the so-called Liouville type formulas for the
number of representations of positive integers by positive diagonal quadratic
forms in nine variables with integral coefficients.
2000 Mathematics Subject Classification: 11E20, 11E25.
Key words and phrases: Entire modular form, congruence subgroup,
theta-functions with characteristics.

In [6], entire modular forms of weight 9/2 on the congruence subgroup Γ0 (4N )
are constructed. The Fourier coefficients of these modular forms have a simple
arithmetical sense. This allows one to get the Liouville type formulas for the
number of representations of positive integers by positive diagonal quadratic

forms in nine variables with integral coefficients.
Let r(n; f ) denote the number of representations of a positive integer n by
the positive quadratic form f = a1 x21 + a2 x22 + · · · + a9 x29 .
1. Preliminaries
1.1. In this paper N , a, d, n, q, r, s denote positive integers; u, v are odd
positive integers; ω is a square free integer; p is a prime number; k, ℓ are nonnegative integers; c, g, h, j, m, x, α, β, γ, δ are integers; z, τ are complex
variables (Im τ > 0); e(z) = exp(2πiz); Q = e(τ ); ( uh ) is the generalized Jacobi
symbol; η(γ) = 1 if γ ≥ 0 and η(γ) = −1 if γ < 0; a denotes the least common
multiple of the coefficients of the quadratic form and ∆ is its determinant.
P
P′
Further,
denote respectively summation with respect to the
and
hmodq

hmodq

complete and the reduced residue system modulo q; ρ(n; f ) is a sum of the
singular series corresponding to r(n; f ).

Since
ϑgh (z | τ ; c, N )
=

X

h(m−c)/N

(−1)

m≡c (mod N )

g
1
m+
e
2N
2
µ

µ

¶2 ¶ µµ

τ e

c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °

g
z
m+
2

¶ ¶

(1.1)

112

G. LOMADZE

(theta-function with characteristics g, h), we have
X
∂
ϑgh (z | τ ; c, N ) = πi
(−1)h(m−c)/N (2m + g)
∂z
m≡c (mod N )

g
1
m+
×e
2N
2
µ

µ

¶2 ¶ µµ

τ e

g
z .
m+
2
¶ ¶

(1.2)

Suppose
ϑgh (τ ; c, N ) = ϑgh (0 | τ ; c, N ),
¯
¯
∂
ϑ′gh (τ ; c, N ) =
ϑgh (z | τ ; c, N )¯¯ .
∂z

z=0

(1.3)

It is known (see, e.g., [5], p.112) that

ϑg+2j,h (τ ; c, N ) = ϑgh (τ ; c + j, N ),

(1.4)

ϑ′g+2j,h (τ ; c, N ) = ϑ′gh (τ ; c + j, N ),
ϑgh (τ ; c + N j, N ) = (−1)hj ϑgh (τ ; c, N ),

(1.5)

ϑ′gh (τ ; c + N j, N ) = (−1)hj ϑ′gh (τ ; c, N ).
From (1.1) and (1.2), in particular according to (1.3), it follows that
ϑgh (τ ; 0, N ) =

∞

X

2 /8N

(−1)hm Q(2N m+g)

m=−∞
∞
X

ϑ′gh (τ ; 0, N ) = πi

,

(−1)hm (2N m + g)Q(2N m+g)

(1.6)
2 /8N

.

(1.7)

m=−∞

From (1.6) and (1.7) we obtain
ϑ−g,h (τ ; 0, N ) = ϑgh (τ ; 0, N ),

ϑ′−g,h (τ ; 0, N ) = −ϑ′gh (τ ; 0, N ).

(1.8)

It is clear that
9
Y

ϑ00 (τ ; 0, 2ak ) = 1 +

∞
X

r(n; f )Qn .

(1.9)

n=1

k=1

Further, put
θ(τ ; f ) = 1 +

∞
X

ρ(n; f )Qn .

(1.10)

n=1

It is known that
ρ(n; f ) =

∞
π 9/2 n7/2 X
A(q),
Γ(9/2)∆1/2 q=1

(1.11)

where Γ(·) is the gamma-function,
A(q) = q

−9/2
h

¶ 9
hn Y
e −

S(ak h, q)
q k=1
mod q

X′

µ

(1.12)

REPRESENTATIONS OF INTEGERS BY THE QUADRATIC FORM

113

and S(ak h, q) is the Gaussian sum.
1.2. For the convenience we quote some known results as the following lemmas.
Lemma 1 ([2], p. 811, the end of the p. 954). The entire modular
form F (τ ) of weight r on the congruence subgroup Γ0 (4N ) is identically zero if
in its expansion in the series
F (τ ) = C0 +

∞
X

Cn Qn ,

n=1
r
12

C0 = Cn = 0 for all n ≤

4N

Q

p|4N

(1 + p1 ).

Lemma 2 ([6], Theorem on p. 64). For given N ≥ a, the function
ψ(τ ; f ) =

9
Y

k=1

ϑ00 (τ ; 0, 2ak ) − θ(τ ; f ) − λ

3
Y

ϑ′gk hk (τ ; 0, 2Nk ),

k=1

where λ is an arbitrary constant, is an entire modular form of weight 9/2, and
the multiplier system
v(M ) = i
Ã

η(γ)(sgn δ−1)/2

·i

(|δ|−1)2 /4

µ

β∆ sgn δ
|δ|

¶

!

α β
on Γ0 (4N ) (M =
is a substitution matrix from Γ0 (4N )) if the following
γ δ
conditions hold:
1) 2 | gk , Nk | N (k = 1, 2, 3), a | N ,
2) 4 | N

3 h2
P
k

k=1

Nk

,4|

3
P
gk2

k=1

4Nk

,

3) for all α and δ with αδ ≡ 1 (mod 4N )
sgn δ

µ

N1 N 2 N3
|δ|

¶Y
3

ϑ′αgk ,hk (τ ; 0, 2Nk ) =

k=1

µ

¶ 3
−∆ Y
ϑ′ (τ ; 0, 2Nk ).
|δ| k=1 gk hk

Lemma 3 (see, e.g., [4], p. 14, Lemma 10). Let
χp = 1 + A(p) + A(p2 ) + · · · ;
then for s > 4
∞
X

q=1

A(q) =

Y

χp .

p

Lemma 4 ([1], Theorem 2, p. 531). Let n be a positive integer and f be
a positive definite quadratic form in s > 4 variables with integral coefficients, ∆

114

G. LOMADZE
′

′

′

its determinant. If 2 ∤ s, 2k || n, 2k || ∆, ∆n = 2k+k uv = r2 ω, pℓ || n, pℓ || ∆,
Q ℓ
Q
′
u=
p = r12 ω1 , v =
pℓ+ℓ = r22 ω2 , then
p|n
p∤2∆

Y
p

p|∆n, p|∆
p>2

¶
µ
Y
Y
(s − 1)!r12−s
s−1
(s−1)/2
1−s −1
χp = s−2 s−1
χ
, (−1)
ω
(1 − p ) L
χ
2 π |Bs−1 | 2 p|∆ p p|2∆
2
p>2

×
×

Yµ

1−

p|r2
p>2

X

d

s−2

µ

(s−1)/2

(−1)

p

¶

p(1−s)/2

¶

(−1)(s−1)/2 ω (1−s)/2
1−
,
p
p

Yµ
p|d

d|r1

ω

µ

¶

¶

where L is the Dirichlet series and Bs−1 is the Bernoulli number.
If in Lemma 4 we put s = 9, then by (1.11) and Lemma 3 we get
Y
Y
8!r1−7
π 9/2 n7/2
χ
(1 − p−8 )−1 L(4, ω)
χ
ρ(n; f ) =
p
2
1/2
7
8
Γ(9/2)∆ 2 π |B8 | p|∆ p|2∆
p>2

×

µ
µ ¶
¶
¶
¶
ω −4 X 7 Y
ω −4
1−
1−
d
p
p
p
p
p|d
d|r1

Yµ

p|r2
p>2

µ

=

Y
Y
n7/2 · 48 · 30
χ
(1 − p−8 )−1 L(4, ω)
χ
p
2
π 4 ∆1/2 r17
p|2∆
p|∆

×

Yµ

p>2

p|r2
p>2

µ
µ ¶
¶
¶
ω −4
ω −4 X 7 Y
1−
.
d
p
p
1−
p
p
p|d
d|r1
¶

µ

(1.13)

Lemma 5 ([3], p. 298). The sum of Dirichlet series L(4, ω) is equal to
11π 4
π4
√ ,
,
L(4,
2)
=
25 · 3
28 · 3 2
¶
X µ h ¶ µ h2
h3
−
L(4, ω) = −2π 4 ω −1/2
ω
22 ω 2 3ω 3
0