Eisenstein series and approximations to pi - Mathematics

12 BRUCE C. BERNDT AND HENG HUAT CHAN
where
1 − J1/n
(4.28) tn := 1 − 3Jn
n √ 1 − Jn
Observe that, by (4.1), (4.2), and Theorem 2.1,
(4.29)
1 − Jn =
8
3Jn
8
3Jn 3 dm2 dJ (Jn,
Jn) .
3 + 1
1/2
Hence, the values of √ 1 − Jn for those n given on page 211 of the Lost Notebook
follow immediately from (2.7). In order to rederive Examples 3.2, it
suffices to compute tn.
Theorem 4.2. If n > 1 is an odd positive integer, then tn lies in the ring
class field of Z[ √ −n].
Proof. By differentiating (4.10) with respect to J(q), we conclude that
(4.30)
dm2 dJ(q) =
1 − J(qn )
G(J(q), J(q
1 − J(q) n )),
for some rational function G(J(q), J(q n )). Using (4.28) and (4.30), we find
that tn can be expressed in terms of Jn. Since Jn is in the ring class field
of Z[ √ −n] when n > 1 is odd and squarefree [14, p. 220, Theorem 11.1], we
complete our proof.
An equivalent form of Theorem 4.2 is first mentioned without proof and
conditions on n by the Chudnovskys on page 391 of [13].
Theorem 4.2 allows us to devise an empirical process for deriving tn whenever
the class group of Q( √ −n) is of the type Zr 2, r ∈ N, where r = 0 refers to
imaginary quadratic fields with class number 1. By (3.6), (4.27), and (4.29),
we find that
(4.31) tn = Qn
Pn − 6
√ .
nπ
Rn
When r = 0, we numerically compute the right hand side of (4.31) using the
definitions of Pn, Qn, and Rn, and then using the command “minpoly(tn,1)”
on the computer algebra system MAPLE V, we derive the values of tn for
n = 3, 7, 11, 19, 27, 43, 67 and 163. We summarize our findings in the following
table.
.