text: special values of zeta function and L-functions
  
  I was reading in some lectures notes about the Riemann zeta-function which takes on special values:
  
  $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$
  
  In fact, we can compute even values of the zeta function $\zeta(2k) \in \pi^{2k}\mathbb{Q}$.
  
  We can twist with a quadratic character and get more special values that way:
  
  $$L(\chi_4, 2k+1) = \sum_{n \geq 0} \frac{\chi_4(n)}{n^{2k+1}} \in \pi^{2k+1}\mathbb{Q}$$
  
  $(\mathbb{Z}/4\mathbb{Z})^* \simeq \mathbb{Z}/2\mathbb{Z}$ so there are two Dirichlet characters mod 4 taking the values $$\begin{array}{c|r|r|r|r|} & 0 & 1 & 2 & 3 \\ \hline \psi & 0 & 1 & 0 & 1 \\ \hline \chi & 0 & -1 & 0 & -1 \\ \hline \end{array}$$
  
  These two examples seem to be part of more general phenomenon. If for dirichlet characters $\psi, \chi$ we fix values at -1: $\chi(-1)=-1$ and if $\psi(-1)=1$, then $$L(2k, \chi) \in \pi^{2k}\sqrt{d} \,\mathbb{Q} \hspace{0.25in}\text{ and }\hspace{0.25in}L(2k+1, \chi) \in \pi^{2k+1}\sqrt{d}\, \mathbb{Q}$$
  
  Where can I find a proof of these rationalitiy results for L-functions?
  
  Do we know anything about the rational numbers that arise in these special values?
  In the case of $\zeta(2k), L(\chi_4, 2k+1)$, the rational numbers are related to the Euler and Bernoulli numbers (e.g. in this paper by N. Elkies).
  
  • – user8268 Nov 1 '13 at 18:48
  • See math.stackexchange.com/a/1771242/30 – nguyen quang do Jul 25 '16 at 11:00
explanation: The text extract is a detailed discussion about special values of the Riemann zeta function and L-functions. It includes specific examples and formulas, such as the value of \(\zeta(2)\) and the behavior of L-functions with quadratic characters. The extract also mentions a more general phenomenon and asks where one can find a proof of the rationality results for L-functions. Additionally, it inquires about the nature of the rational numbers that arise in these special values, particularly their relation to Euler and Bernoulli numbers.
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distilabel_metadata.content: special values of zeta function and L-functions
  
  I was reading in some lectures notes about the Riemann zeta-function which takes on special values:
  
  $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$
  
  In fact, we can compute even values of the zeta function $\zeta(2k) \in \pi^{2k}\mathbb{Q}$.
  
  We can twist with a quadratic character and get more special values that way:
  
  $$L(\chi_4, 2k+1) = \sum_{n \geq 0} \frac{\chi_4(n)}{n^{2k+1}} \in \pi^{2k+1}\mathbb{Q}$$
  
  $(\mathbb{Z}/4\mathbb{Z})^* \simeq \mathbb{Z}/2\mathbb{Z}$ so there are two Dirichlet characters mod 4 taking the values $$\begin{array}{c|r|r|r|r|} & 0 & 1 & 2 & 3 \\ \hline \psi & 0 & 1 & 0 & 1 \\ \hline \chi & 0 & -1 & 0 & -1 \\ \hline \end{array}$$
  
  These two examples seem to be part of more general phenomenon. If for dirichlet characters $\psi, \chi$ we fix values at -1: $\chi(-1)=-1$ and if $\psi(-1)=1$, then $$L(2k, \chi) \in \pi^{2k}\sqrt{d} \,\mathbb{Q} \hspace{0.25in}\text{ and }\hspace{0.25in}L(2k+1, \chi) \in \pi^{2k+1}\sqrt{d}\, \mathbb{Q}$$
  
  Where can I find a proof of these rationalitiy results for L-functions?
  
  Do we know anything about the rational numbers that arise in these special values?
  In the case of $\zeta(2k), L(\chi_4, 2k+1)$, the rational numbers are related to the Euler and Bernoulli numbers (e.g. in this paper by N. Elkies).
  
  • – user8268 Nov 1 '13 at 18:48
  • See math.stackexchange.com/a/1771242/30 – nguyen quang do Jul 25 '16 at 11:00
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a detailed discussion about special values of the Riemann zeta function and L-functions. It includes specific examples and formulas, such as the value of \(\zeta(2)\) and the behavior of L-functions with quadratic characters. The extract also mentions a more general phenomenon and asks where one can find a proof of the rationality results for L-functions. Additionally, it inquires about the nature of the rational numbers that arise in these special values, particularly their relation to Euler and Bernoulli numbers.
  
  Label: question-and-answer