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
The text passage discusses special values of the Riemann zeta function and L-functions, providing specific examples and mentioning their connections to Euler and Bernoulli numbers. It also asks for references to proofs of rationality results for L-functions and inquires about the nature of the rational numbers that arise in these special values.
multiple_questions

1. "where can i find a proof of these rationality results for l-functions?"
2. "do we know anything about the rational numbers that arise in these special values?"
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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
user
The text passage discusses special values of the Riemann zeta function and L-functions, providing specific examples and mentioning their connections to Euler and Bernoulli numbers. It also asks for references to proofs of rationality results for L-functions and inquires about the nature of the rational numbers that arise in these special values.

TYPE: MULTIPLE_QUESTIONS

1. "Where can I find a proof of these rationality results for L-functions?"
2. "Do we know anything about the rational numbers that arise in these special values?"