What does $L(n,\chi_4)$ mean?

Question

I was reading some articles related to Euler sums and the Riemann zeta function, when I came across this definition:

$$ L(n,\chi_4) = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^n} $$

What is this function called and how is it related to the zeta function?

$\chi_4(n) = (-1)^{(n-1)/2} 1_{2 |n-1}$ is the unique non-trivial Dirichlet character modulo $4$. It is completely multiplicative so that $L(s,\chi_4) = \sum_{n=1}^\infty \chi(n) n^{-s}=\prod_p \frac{1}{1-\chi_4(p) p^{-s}}$. Also $L(s,\chi_4)$ is meromorphic (and even entire) on the whole complex plane, and since $\chi_4$ is its own discrete Fourier transform, it has a functional equation similar to $\zeta(s)$. In one word, a Riemann hypothesis is conjectured for $L(s,\chi_4)$. Those Dirichlet L-functions are also the core of the proof of Dirichlet theorem in arithmetic progressions — reuns, Jul 21 '17 at 21:52
@Pickle You may also be interested by this parallel exposition of the explicit formulas for $\zeta$ and your Dirichlet L-function with a link to a table of the imaginary parts of the first zeros. — Raymond Manzoni, Jul 22 '17 at 07:34
@Pickle: your specific series is named a Dirichlet beta function and allows (for $n$ integer) a parallel between Euler and Bernoulli numbers. — Raymond Manzoni, Jul 23 '17 at 11:41
@RaymondManzoni Thanks a lot! I was actually wondering if there was a specific name for the case where the character is $\chi_4$. That's exactly what I was looking for! — Klangen, Jul 24 '17 at 08:46
@RaymondManzoni The sum in the Dirichlet Beta function goes from 0 to infinity, however here it goes from 1 to infinty... — Klangen, Jul 24 '17 at 08:47
Thanks to appreciate my links @Pickle! Should you reference my last link : the OP starts with $n=1$ but with $(2n-1)$ instead of $(2n+1)$ (with a wrong overall sign as commented by anon). Excellent continuation anyway, — Raymond Manzoni, Jul 24 '17 at 22:47

GEdgar · Accepted Answer · 2017-07-21T21:15:10.087

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That author is writing $\chi_4$ for a certain "character", defined by $$ \chi_4(n) = \begin{cases} 1, &n \equiv 1\pmod 4,\\ -1,&n\equiv 3\pmod 4,\\ 0, &\text{otherwise} \end{cases} $$ and then $$ L(s,\chi_4) = \sum_{n=1}^\infty \frac{\chi_4(n)}{n^s} = \sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^s} $$ is the corresponding "$L$-function" of that character.

The simple connection with the zeta function is that, taking a different character $\chi(n) = 1$ for all $n$, we get $$ L(s,\chi) = \sum_{n=1}^{\infty}\frac{1}{n^s} = \zeta(s) $$

edited Jul 21 '17 at 21:15

answered Jul 21 '17 at 21:09

GEdgar

111,679

Thank you for the answer. However, it is not very clear to me how the denominator goes from $n^s$ to $(2k+1)^s$ when applying $\chi_4$ to the numerator... – Klangen Jul 24 '17 at 08:45
1

$\chi_4(n)=0$ for even $n$, so leave those terms out of the sum. For odd $n$, write $n = 2k+1$ so $\chi_4(2k+1)=(-1)^k$. – GEdgar Jul 24 '17 at 08:56
Perfect, thanks a lot! – Klangen Jul 24 '17 at 08:58

What does $L(n,\chi_4)$ mean?

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