Difference between revisions of "Riemann zeta at even integers"
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(Created page with "==Theorem== The following formula holds for even integers $n$ and $m \in \{1,2,3,\ldots\}$: $$\zeta(n)= \left\{ \begin{array}{ll} 0 &, \quad n=-2m, \\ -\dfrac{1}{2} &, \quad n...") |
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==References== | ==References== | ||
| − | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta as contour integral|next= | + | * {{BookReference|The Zeta-Function of Riemann|1930|Edward Charles Titchmarsh|prev=Riemann zeta as contour integral|next=Functional equation for Riemann zeta}}: § Introduction $(5)$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 23:57, 17 March 2017
Theorem
The following formula holds for even integers $n$ and $m \in \{1,2,3,\ldots\}$: $$\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
- 1930: Edward Charles Titchmarsh: The Zeta-Function of Riemann ... (previous) ... (next): § Introduction $(5)$
