Riemann zeta at even integers

From specialfunctionswiki
Revision as of 23:45, 17 March 2017 by Tom (talk | contribs) (Created page with "==Theorem== The following formula holds for even integers $n$ and $m \in \{1,2,3,\ldots\}$: $$\zeta(n)= \left\{ \begin{array}{ll} 0 &, \quad n=-2m, \\ -\dfrac{1}{2} &, \quad n...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Theorem

The following formula holds for even integers $n$ and $m \in \{1,2,3,\ldots\}$: $$\zeta(n)= \left\{ \begin{array}{ll} 0 &, \quad n=-2m, \\ -\dfrac{1}{2} &, \quad n=0 \\ \dfrac{(-1)^m B_m}{2m} &, \quad n=2m, \end{array} \right.$$ where $\zeta$ denotes Riemann zeta and $B_m$ denotes Bernoulli numbers.

Proof

References