new mathematical object found

galactus · #1 $Old$ April 15th, 2008, 06:26 PM

Here is an interesting article.

Math Gateway

And another:

Math Gateway

topsquark · #2 $Old$ June 2nd, 2008, 07:38 PM

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

(I evidently missed this when it was posted.)

-Dan

Jhevon · #3 $Old$ June 2nd, 2008, 08:01 PM

Quote:

Originally Posted by galactus $View Post$

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?!

ThePerfectHacker · #4 $Old$ June 2nd, 2008, 08:20 PM

Quote:

Originally Posted by topsquark $View Post$

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

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

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