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II. The Partition Function
24
and we try C a circle near the unit circle, i.e.,
C is |z| r, r < 1. à 13)
Next we break up C as dictated by our consideration of |1−z| ,
1−|z||
namely, into
|1 − z|
A is the arc |z| r, ≤ 3,
1 −|z|
and (14)

|1 − z|
B is the arc |z| r, ≥ 3.
1 −|z|
So,
p(n) − q(n) (15)

1 Fàz) − φ(z) 1 Fàz) − φ(z)
dz + dz,
2πi A z n+1 2πi B z n+1
and if we use (7) on this first integral and (9), (11) on this second
integral we derive the following estimates:

1 Fàz) − φ(z)
dz
2πi A z n+1

M π 2 1
(1 − r) 3/2 exp × the length of A.
r n+1 6 1 − r

(M is the implied constant in the O of (7) when c 3).
As for the length of A, elementary geometry gives the formula
√
2(1 − r)
4r arcsin √
r
and this is easily seen to be O(1 − r). We finally obtain, then,

1 Fàz) − φ(z)
dz
2πi A z n+1

(1 − r) 5/2 . π 2 1
M exp , à 16)
r n 6 1 − r
where M is an absolute constant.