In this section we briefly describe one way to evaluate ,
for real. See [Dok04] for a more sophisticated
analysis of computing and its Taylor expansion for any
complex number .
Theorem 1.10 (Lavrik)
We have the following
rapidly-converging series expression for , for any
complex number :
where
and
is the incomplete -function.
Theorem 1.10 above is a special case of a more
general theorem that gives rapidly converging series
that allow computation of any Dirichlet series
that meromorphically continues to the whole complex plane and
satisfies an appropriate functional equation.
For more details, see [Coh00, §10.3],
especially Exercise 24 on page 521 of [Coh00].
Subsections
William
2007-05-25