Talk:Dirichlet character

message: Historically, what does the L in L-function or L-series stand for? Lagrange? Ninte 11:40, 13 July 2006 (UTC)
thread_title:  History 

message: It seems obvious to me that the following tables are missing, but since I know nothing of this subject beyond what I've just read here I better check:
:=== Modulus 1 ===
:There are \phi(1)=1 characters modulo 1:
:
:
χ \ n  	1
\chi_1(n)                      	1
:
:=== Modulus 2 ===
:There are \phi(2)=1 characters modulo 2:
:
χ \ n  	1	2
\chi_1(n)                      	1	0
Since the text claims to present "all of the characters up to modulus 7", these should be included, no? Hv 15:52, 9 February 2006 (UTC)
replies: text: Yes, that is correct. But perhaps these can continue to be left out, for, I fear, they would contribute to a mind-numbing clutter? Sometimes, less is more. linas 00:43, 10 February 2006 (UTC)
replies: text: If it is clutter, maybe it'd be better to show only the table for 7 - I think the rest are pretty obvious once you understand those. I've corrected the comment at the top to promise only what is actually there. Hv 11:18, 10 February 2006 (UTC)
thread_title:  Missing character tables 

message: I am not an expert on Dirichlet characters, but ... are we sure that the first and third conditions really are sufficient? - Doesn't the identically zero function satisfy the first and third conditions, whilst not being a Dirichlet character? Madmath789 21:21, 26 June 2006 (UTC)
replies: text: It appears someone cleaned up the article already linas 04:04, 9 July 2006 (UTC)
thread_title: Sufficient conditions

message: Just wondering something... of the two mod 4 characters on this page, the first {1,0,1,0} seems to be the same as the sole mod 2 character {1,0}. Does it really need to be in the list twice? I think it should be deleted from the mod 4 list, considering its period is 2; that would leave the sole mod 4 character as {1,0,-1,0}.
Bird of paradox 09:34, 29 September 2006 (UTC)
replies: text: I don't think it would be correct to remove it, as that would make it appear as if there were really only one character mod 4. There is a theorem about the number of characters mod n, and it tells us that there are 2 mod 4 characters - it just happens that one of them is the same as the only mod 2 character. Madmath789 09:41, 29 September 2006 (UTC)
thread_title:  mod 4 character 

message: There was an error in this table which I have corrected. The error set \omega = \exp(\pi i /6) but as a circle is 2 \pi radians this should be \omega = \exp(\pi i /3) Wilmot1 08:04, 19 July 2007 (UTC)
thread_title:  mod 7 character 

message: This is how I would try to introduce the concept :
Consider the field \mathbb{Z} / p \mathbb{Z} where p is prime. the multiplicative group G = (\mathbb{Z} / p \mathbb{Z},\times) can be mapped to the additive group (\mathbb{Z} / (p-1)\mathbb{Z},+) : if g is a generator of G then  every element of G can be uniquely written as a power of g, the exponent being an element of (\mathbb{Z} / (p-1)\mathbb{Z},+). 
The Dirichlet character exploits this idea : 
*choose a prime p which will be the periodicity of the character \chi(n+p) = \chi(n)
*set \chi(n) = 0 if n \equiv 0 \pmod{p},
*choose a generator of G = (\mathbb{Z} / p\mathbb{Z},\times), and a p-1 root of unity \omega = e^{2 i k \pi / (p-1)}
*for n \not\equiv 0 \pmod{p} set \chi(n) = \omega^a where a = \log_g(n) the discrete logarithm modulo p , i.e. n \equiv g^a \pmod{p}. 
The same idea can be extended to characters of non-prime periodicity. After that, we remark that a change of generator can be seen as a change of root of the unity, so that all the Dirichlet characters of periodicity p can be constructed from the same generator but with different roots of the unity. And this is how we get to the discrete Fourier transform (of size \varphi(p), i.e p-1 if p if prime) and how we prove that the Dirichlet character of a same periodicity are all orthogonal. Because two integers always have a lcm, two Dirichlet characters can always thought as being of the same periodicity, and thus are mutually orthogonal. So that all the Dirichlet characters together form an orthogonal basis of the vector space of periodic sequences (u_n)_{n \in \mathbb{N}^*}. Then, the Euler product of \sum_n \frac{\chi(n)}{n^s} = \prod_p \frac{1}{1 - \chi(p) p^{-s}} is exhibited.
And finally by noting Y(k) = \sum_{n \in G} \chi(n) e^{2 i \pi n k / p} the discrete fourier transform of \chi we get that \forall a \in G, \quad\sum_{n \in G} \chi(an) e^{2 i \pi a n k / p} = \chi(a) Y(k) so that Y(k) = Y(1) \bar{\chi}(n). from that we can write a Fourier series representation of \sum_n \chi(n) \delta(x-n) = \frac{1}{q} \sum_{k=1}^\infty \bar{\chi}(k)\left(Y(1) e^{-2 i \pi k x / q} + Y(-1) e^{2 i \pi k x / q}\right) which leads to the functional equation L(\chi,s) = \sum_{n=1}^\infty \chi(n) n^{-s} = \int_0^\infty \sum_n \chi(n) \delta(x-n) x^{-s} dx  = \sum_{k=1}^\infty \bar{\chi}(k) k^{s-1} \frac{1}{q}\int_0^\infty \left(Y(1) e^{-2 i \pi k x / q} + Y(-1) e^{2 i \pi k x / q}-2\right) x^{-s}dx = L(1-s,\chi) A(s)
Dirichlet characters are thus a nice mathematical construction based on no less than : 
*simple arithmetic, 
*group theory, 
*linear algebra, 
*complex analysis, Fourier analysis, Dirichlet series, Mellin/Laplace/Fourier transforms, zeta functions / L-functions...
78.227.78.135 (talk) 20:12, 2 January 2016 (UTC)
thread_title:  Simple explanation 

message: Riemann might have extended the original definition of the Zeta function. I am not sure that he referred to all the L(s,chi) functions.  — Preceding unsigned comment added by 2A00:23C4:7C87:4F00:15D4:7730:972E:49E (talk) 11:54, 8 July 2020 (UTC)
thread_title:  Extension 

message: I used to edit math articles under the name Virginia-American but my computer died, etc., and I just recently got back.
I completely rewrote this article. Unfortunately, I submitted it for review as though it were a new article, so it's not on my sandbox page  but is here: Draft:Dirichlet character.
I think my rewrite  is a big improvement, but I'd like some feedback before I post it.
Thanks
James in dc (talk) 21:43, 13 June 2021 (UTC)
thread_title:  Proposed Rewrite 

message: Done.
"Be bold!"
James in dc (talk) 21:11, 30 June 2021 (UTC)
thread_title:  Rewritten 

message: My rewrite has been posted for a month. No comments, corrections, or anything from onyone
And now it's reverted.
James in dc (talk) 14:06, 1 August 2021 (UTC)
thread_title:  Vandalism 

message: Two weeks later, I'm reposting my rewrite.
If anyone has anything constructive (errors, additions, whatever) to say, say it.
And, could someone more knowledgeable than me post categories and rankings
Thanks
James in dc (talk) 15:36, 18 August 2021 (UTC)
thread_title:  Undo Vandalism