As I promised in my previous post, here is a derivation of the analytic continuation of the Riemann zeta function to negative integer values. There are several ways of doing this but a particularly simple way is given by Graham Everest, Christian Rottger, and Tom Ward at this link. It starts with the observation that you can write
if the real part of
For
, you can expand the integrand in a binomial expansion (2)Now substitute (2) into (1) to obtain
(3)or
(3′)where the remainder
is an analytic function whenwhich implies that
(4)Taking the limit of
going to zero from the right of (3′) givesHence, the analytic continuation of the zeta function to zero is -1/2.
The analytic domain of
can be pushed further into the left hand plane by extending the binomial expansion in (2) toInserting into (1) yields
where
is analytic for(5)
Rearranging (5) gives
(6)where I have used
The righthand side of (6) is now defined for
Collecting terms, substituting for
and multiplying by givesReindexing gives
Now, note that the Bernoulli numbers satisfy the condition
. Hence, letand obtain
which using
and gives the self-consistent condition,which is the analytic continuation of the zeta function for integers
.