# Riemann Zeta Function at Non-Positive Integers

## Theorem

Let $n \ge 0$ be a integer.

Then:

$\map \zeta {-n} = \paren {-1}^n \dfrac {B_{n + 1} } {n + 1}$

where:

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

## Proof

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

where $C$ is the Hankel contour.

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

So:

 $\displaystyle \map \zeta {-n}$ $=$ $\displaystyle \frac {i n!} {2 \pi} \oint_C \frac 1 {\paren {-z}^{n + 2} } \cdot \frac {-z} {e^z - 1} \rd z$ Gamma Function Extends Factorial $\displaystyle$ $=$ $\displaystyle -\frac {i n!} {2 \pi} \cdot 2 \pi i \Res {\frac 1 {\paren {-z}^{n + 2} } \cdot \frac {-z} {e^z - 1} } 0$ Residue Theorem $\displaystyle$ $=$ $\displaystyle n! \Res {\paren {-1}^2 \paren {-1}^n \sum _{k \mathop = 0}^\infty \frac {B_k} {k!} z^{k - n - 2} } 0$ Definition of Bernoulli Numbers, $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle n! \paren {-1}^n \Res {\sum _{k \mathop = 0}^\infty \frac {B_k} {k!} z^{k - n - 2} } 0$

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:

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

$\blacksquare$