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 . 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 because the resulting series is absolutely convergent. Since the zeta function is analytic for , the right hand side is a new definition of that is analytic for 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 . 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 . 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 .