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VI. A “Natural” Proof of the Nonvanishing of L-Series
64
θt
e
Since
t
e −1 − e −t is analytic and has integrable derivatives on [0, ∞),
t
we may integrate by parts repeatedly and thereby get
1 1
−
(n − θ) z z − 1

1 ∞ d k e θt e −t
− − t z+k−1 dt.
t
A(z + kð 0 dt e − 1 t
This gives continuation to z> −k, and, since k is arbitrary, the
continuation is to the entire plane.