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

By Hankel Representation of Riemann Zeta Function:

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

where $C$ is the Hankel contour.

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

So:

\(\ds \map \zeta {-n}\)

\(=\)

\(\ds \frac {i n!} {2 \pi} \oint_C \frac 1 {\paren {-z}^{n + 2} } \cdot \frac {-z} {e^z - 1} \rd z\)

Gamma Function Extends Factorial

\(\ds \)

\(=\)

\(\ds -\frac {i n!} {2 \pi} \cdot 2 \pi i \, \Res {\frac 1 {\paren {-z}^{n + 2} } \cdot \frac {-z} {e^z - 1} } 0\)

Cauchy's Residue Theorem

\(\ds \)

\(=\)

\(\ds n! \, \Res {\paren {-1}^2 \paren {-1}^n \sum _{k \mathop = 0}^\infty \frac {B_k} {k!} z^{k - n - 2} } 0\)

Definition of Bernoulli Numbers, $i^2 = -1$

\(\ds \)

\(=\)

\(\ds n! \paren {-1}^n \, \Res {\sum_{k \mathop = 0}^\infty \frac {B_k} {k!} z^{k - n - 2} } 0\)

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:

\(\ds \map \zeta {-n}\)

\(=\)

\(\ds n! \paren {-1}^n \frac {B_{n + 1} } {\paren {n + 1}!}\)

\(\ds \)

\(=\)

\(\ds \paren {-1}^n \frac {B_{n + 1} } {n + 1}\)

$\blacksquare$

Examples

Example: $\map \zeta {-1}$

$\map \zeta {-1 } = -\dfrac 1 {12} $

Example: $\map \zeta {-3}$

$\map \zeta {-3 } = \dfrac 1 {120} $