1.4. DEDEKIND’S ETA-FUNCTION 17

Definition 1.58. Dedekind’s eta-function, denoted by η(z), is defined by the

infinite product

η(z) :=

q1/24

∞

n=1

(1 −

qn).

Remark 1.59. By this definition, it is obvious that η(z) is nonvanishing on H.

The eta-function is useful for providing explicit descriptions of many modular

forms, and is also useful for constructing combinatorial generating functions. Using

Jacobi’s Triple Product Identity (see Theorem 2.8 of [And6])

(1.26)

∞

n=1

(1 −

x2n)(1

+

x2n−1z2)(1

+

x2n−1z−2)

=

∞

m=−∞

z2mxm2

,

it is not diﬃcult to derive the following classical q-series identities for certain theta-

functions.

Theorem 1.60. The following q-series identities are true:

η(24z) = q

∞

n=1

(1 −

q24n)

=

∞

k=−∞

(−1)kq(6k+1)2

=

1

2

θ

12

•

, 0, z ,

η(8z)3

= q

∞

n=1

(1 −

q8n)3

=

∞

k=0

(−1)k(2k

+

1)q(2k+1)2

= θ

−4

•

, 1, z ,

η(z)2

η(2z)

=

∞

n=1

(1 −

qn)2

(1 − q2n)

=

∞

n=−∞

(−1)nqn2

,

η(2z)5

η(z)2η(4z)2

=

∞

n=1

(1 − q2n)5

(1 − qn)2(1 − q4n)2

= θ0(z) =

∞

n=−∞

qn2

,

η(16z)2

η(8z)

= q

∞

n=1

(1 −

q16n)2

(1 − q8n)

=

∞

n=0

q(2n+1)2

.

By the first identity in Theorem 1.60, it is obvious that Dedekind’s eta-function

is a modular form of weight 1/2. More precisely, we have the following description

of its modular transformation properties (see, for example, Theorem 3.1 of [Apo]).

Theorem 1.61. For z ∈ H, we have

η(z + 1) =

eπi/12η(z),

η(−1/z) =

(−iz)1/2η(z).

Using Theorem 1.61 and the definition of half-integral weight modular forms, it is

not diﬃcult to obtain a second deduction of the following fact.

Corollary 1.62. We have that

η(24z) ∈ S

1

2

(Γ0(576), χ12),