Welcome to MathOverflow

Question

4

1

I was hoping to see this pop up on the recent big list question about etymology or terms and symbols. Since it has not, and I can't find an answer, I will ask:

What is the reason for the $L$ in $L$-function? I've read that the general use of the term cames from Dirichlet's $L$-functions $L(s,\chi).$ Was there any motivation behind Dirichlet's use or was it just an arbitary choice?

If so, is there any compelling reason that we keep this name other than tradition?

flag	asked Apr 14 2010 at 19:10 Jamie Weigandt 2,123●1●11●20

5

Here's one way of looking at it. Dirichlet had to use some letter. He used L. Whatever he had used---would you have asked what the reason was? Why do number theorists use T for Hecke algebras? It's just what someone chose and it stuck. It might be no more than that... – Kevin Buzzard Apr 14 2010 at 19:37

Did Dirichlet actually use L? – François G. Dorais♦ Apr 14 2010 at 21:34

1

François, Dirichlet absolutely used $L$. Look at his papers on primes in arithmetic progressions. – KConrad Apr 15 2010 at 2:43

Kevin has an excellent point. I just wondered because L-functions are very important tools with a non-descriptive name. I like Paul's retroactive interpretation that L stands for Langlands. – Jamie Weigandt Apr 16 2010 at 21:39

KConrad · Accepted Answer · 2010-04-14 22:09:39Z UTC

It is not known why Dirichlet denoted his functions with an $L$. Perhaps he chose $L$ for Legendre (I am not serious). The reason may be alphabetical. Just before $L$-functions are introduced in his 1837 paper on primes in arithmetic progression (Math. Werke vol. 1, 313--342), there are certain functions $G$ and $H$, and the letters $I, J$, and $K$ may not have seemed appropriate labels for a function.

While $L(s,\chi)$ and $L(\chi,s)$ are common notations for the $L$-function of a character $\chi$, neither decorated notation is due to Dirichlet; he simply wrote different $L$-functions as $L_0, L_1, L_2,\dots$.

Barry · Answer 2 · 2010-04-14 19:23:15Z UTC

5

Many have suggested that it comes from "Lejeune", as in "Johann Peter Gustav Lejeune Dirichlet". I have never seen this properly sourced and have often wondered if the claim is legitimate.

answered Apr 14 2010 at 19:23

Barry
675●6●11

3

I have heard that as well, but I always thought it was mostly a humorous way to say "We have no idea where the terminology actually comes from". – Pete L. Clark Apr 14 2010 at 22:56

1

I heard that Dirichlet had some disdain of his Walloon heritage. If so, I doubt that he would decide to "honor" the part of his name that most reflects that... – François G. Dorais♦ Apr 14 2010 at 23:59

Paul Broussous · Answer 3 · 2010-04-15 07:26:09Z UTC

Whatever the historical reasons are, I think it is a good thing to use the terminology 'L-function' because of Langlands's amazing contribution to the theory of automorphic forms. Moreover Langlands functorialities are stated in terms of the 'L-group'.

Why are they called L-functions?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.