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),
Previous Page Next Page