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 0" />0" title="Analytic continuation continued" />0" class="latex" title="s>0" />. You can then break the integral into pieces with
(1)
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 when -1" /> -1" title="Analytic continuation continued" /> -1" class="latex" title="Re s > -1" /> because the resulting series is absolutely convergent. Since the zeta function is analytic for 1" />1" title="Analytic continuation continued" />1" class="latex" title="Re s >1" />, the right hand side is a new definition of that is analytic for 0" />0" title="Analytic continuation continued" />0" class="latex" title="s >0" /> aside from a simple pole at . Now multiply (3) by and take the limit as to obtain
which implies that
(4)
Taking the limit of going to zero from the right of (3′) gives
Hence, 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) to
Inserting into (1) yields
where is analytic for -(k+1)" />-(k+1)" title="Analytic continuation continued" />-(k+1)" class="latex" title="Re s>-(k+1)" />. Now let and extract out the last term of the sum with (4) to obtain
(5)
Rearranging (5) gives
(6)
where I have used
The righthand side of (6) is now defined for -k" /> -k" title="Analytic continuation continued" /> -k" class="latex" title="Re s > -k" />. Rewrite (6) as
Collecting terms, substituting for and multiplying by gives
Reindexing gives
Now, note that the Bernoulli numbers satisfy the condition . Hence, let
and obtain
which using and gives the self-consistent condition
,
which is the analytic continuation of the zeta function for integers .