Riemann Zeta Function at Non-Positive Integers

Theorem
Let $n \ge 0$ be a integer.

Then:
 * $\displaystyle \map \zeta {-n} = \paren {-1}^n \frac{ B_{n + 1} } {n + 1}$

where:
 * $B_n$ is the $n$th Bernoulli number
 * $\zeta$ is the Riemann Zeta function

Proof
By Hankel Representation of Riemann Zeta Function:


 * $\displaystyle \map \zeta {-n} = \frac {i \map \Gamma {1 + n} } {2 \pi} \oint_C \frac 1 {z^{n + 1} \paren {e^z - 1} } \rd z$

where $C$ is the Hankel contour.

Note that the integrand is meromorphic, with a pole at $z = 0$ lying inside the contour.

So:

By the definition of a residue, the residue at $0$ is given by the coefficient of the $\dfrac 1 z$ term.

This is the term where $k = n + 1$, so: