Cryptography Reference
In-Depth Information
Using the facts that
ζ
(4) =
π
4
90
π
6
945
,
and
ζ
(6) =
we obtain
⎛
⎝
1 + 240
∞
⎞
g
2
(
τ
)=
4
π
4
(1 + 240
q
+
···
)=
4
π
4
3
j
3
q
j
1
− q
j
⎠
3
j
=1
⎛
⎞
504
∞
g
3
(
τ
)=
8
π
6
)=
8
π
6
27
j
5
q
j
1
− q
j
⎝
1
⎠
.
27
(1
−
504
q
+
···
−
j
=1
Since Δ =
g
2
−
27
g
3
, a straightforward calculation shows that
Δ(
τ
)=(2
π
)
12
(
q
+
···
)
.
Define
j
(
τ
) = 1728
g
2
Δ
.
Then
j
(
τ
)=
q
+
···
. Including a few more terms in the above calculations
yields
j
(
τ
)=
1
q
+ 744 + 196884
q
+ 21493760
q
2
+
···
.
For computational purposes, this series converges slowly since the coe
cients
are large. It is usually better to use the following.
PROPOSITION 9.12
1 + 240
j
=1
q
j
3
j
3
q
j
1
−
j
(
τ
) = 1728
q
j
2
.
1 + 240
j
=1
q
j
3
1
−
504
j
=1
j
3
q
j
1
j
5
q
j
1
−
−
−
PROOF
Substitute the above expressions for
g
2
,g
3
into the definition of
the
j
-function. The powers of
π
and other constants cancel to yield the present
expression.
It can be shown (see [70, p. 249]) that
Δ=(2
π
)
12
q
∞
(1
− q
k
)
24
.
k
=1
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