# Thread: new mathematical object found

1. ## new mathematical object found

Here is an interesting article.

Math Gateway

And another:

Math Gateway

2. All right. I'll bite. What's an L-function?

(I evidently missed this when it was posted.)

-Dan

3. Originally Posted by galactus
Here is an interesting article.

Math Gateway

And another:

Math Gateway
They called me crazy! but i always knew it existed! Not so crazy now, huh? Huh?!

4. Originally Posted by topsquark
All right. I'll bite. What's an L-function?

(I evidently missed this when it was posted.)

-Dan
It is a thing in number theory.

A Dirichlet charachter is a function $\displaystyle \chi : \mathbb{F}_p^{\times}\mapsto \mathbb{C}^{\times}$ (here $\displaystyle \mathbb{F}_p^{\times}$ is the non-zero elements mod $\displaystyle p$ and $\displaystyle \mathbb{C}^{\times}$ are the non-zero complex numbers) which satisfies $\displaystyle \chi (ab) = \chi (a)\chi (b)$.

Given a Dirichlet charachter $\displaystyle \chi$ we form the Dirichlet L-series $\displaystyle L(\chi , s)= \sum_{n=1}^{\infty}\frac{\chi (n)}{n^s}$ where $\displaystyle s$ is complex.

It can be proven that $\displaystyle L(\chi , s)$ converges when $\displaystyle \Re (s) > 1$.
Using analytic continuation we extend this to a meromorphic function on $\displaystyle \mathbb{C}$.
This continued new function is called the Dirichlet L-function.
It is related to Riemann zeta function.