Talk:Dirichlet character

History

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

Missing character tables

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)

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)

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)

Sufficient conditions

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)

It appears someone cleaned up the article already linas 04:04, 9 July 2006 (UTC)

mod 4 character

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)

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)

mod 7 character

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)

There is actually a mistake in this table. Consider $\chi _{2}$. Then $\chi _{2}(2)=\chi _{2}(3)^{2}=(\omega ^{2})^{2}=\omega ^{4}=-\omega$, but the table says $\chi _{2}(2)=-\omega ^{2}$. This makes me wonder how many other mistakes are in the tables for the Dirichlet characters; these should all be carefully checked. — Preceding unsigned comment added by 31.125.12.109 (talk) 11:43, 21 August 2020 (UTC)

Definitions of primitivity

I wrote An equivalent definition of a primitive Dirichlet character is to consider the associated multiplicative character and define it to be primitive if the period of the character is exactly the modulus. and cited Davenport (1967). I didn't give the page number, but it's page 37. User:EmilJ deleted that with the edit summary "This is wrong. For example, the principal character of square-free modulus n > 1 has period n, but it is not primitive." I assume it's me rather than Davenport who was wrong here? Anyway, the source says "It is possible however that for values of n restricted by the condition (n,q)=1 the function χ(n) may have a period less than q. If so, we say that χ is imprimitive and otherwise primitive". I think that what I wrote is a fair rendition of that, and if not I invite improvements. However, EmilJ's comment is wrong in itself. The principal character has as its associated character the multiplicative function which is always 1, and this has period 1. So by my rendition it is not primitive. Deltahedron (talk) 20:09, 30 August 2012 (UTC)

I think the terminology you've used is non-standard and has lead to confusion. By "the associated multiplicative character", I assume you mean the character of (Z/nZ)^×. Then, you are correct, but EmilJ probably thought you meant the multiplicative function on the integers. RobHar (talk) 01:41, 31 August 2012 (UTC)

Good point, perhaps "character on the multiplicative group" would have been better. How about consider the associated character on multiplicative groups and define it to be primitive if the character is not defined on any smaller modulus. Deltahedron (talk) 06:02, 31 August 2012 (UTC)

The principal character mod n is the multiplicative function which is 1 for k such that gcd(n,k), and 0 otherwise. This function does not have period 1. As RobHar wrote, your terminology is confusing. If you really meant a character of (Z/nZ)^×, this should be written explicitly, but even then it is confusing as one usually speaks about periods for functions defined on integers, not functions defined on finite groups, and the definition of period in such a case would require an explanation. The quoted description by Davenport is yet different: he considers the character as a function on the subset $\{q\in \mathbb {Z} :(n,q)=1\}$ of integers, which is presumably meant to have period m if $\chi (k+am)=\chi (k)$ whenever both sides are defined. Then a character is indeed imprimitive if it has a period smaller than n. Note the indefinite article: a periodic function does not have a unique period, any multiple will do, and in particular, a character mod n always has period n. Only the minimal period (= conductor) of an imprimitive character (in Davenport’s sense) is guaranteed to be less than n.—Emil J. 10:25, 31 August 2012 (UTC)

The character on the multiplicative group Z/n associated with the principal Dirichlet character is the character which always takes the value 1. This is defined on a smaller modulus, namely 1. (As it happens, Davenport at p.37 thinks it "a matter of personal preference whether one includes the principal character among the imprimitive characters, I prefer to leave it unclassified".) I already suggested an improved wording. Deltahedron (talk) 16:37, 31 August 2012 (UTC)

I also have a quibble about the statement The imprimitive characters correspond to missing Euler factors in the associated L-functions. This is sensitive to the choice of what kind of characters one considers. For example, let χ be the nonprincipal character mod 4, and ψ be the induced character mod 8. Then L(χ,s) = L(ψ,s), so no Euler factor is missing. This is not a problem if we define characters purely as completely multiplicative periodic functions on the integers, since then χ and ψ are identical characters. However, if we either consider a specification of a modulus as a part of the definition of a character, or consider characters on (Z/nZ)^×, then the two characters are distinct, and ψ is imprimitive. Presumably this should be clarified.—Emil J. 11:14, 31 August 2012 (UTC)

The article said Imprimitive characters can cause missing Euler factors in L-functions and I changed it to The imprimitive characters correspond to missing Euler factors in the associated L-functions. Davenport (p.39) says "The relation (2) between an imprimitive character and the primitive character which induces it implies a simple relation between the corresponding L-functions" and follows with a formula involving the Euler factors at primes dividing the modulus of the imprimitive factor. Apostol (p.262) says "every L-series is equal to the L-series of a primitive character, multiplied by a finite number of factors". Suggestions for a simpler summary would be welcome. Deltahedron (talk) 16:49, 31 August 2012 (UTC)

The original wording is in fact fine. The “can” takes care of the problem that not every imprimitive characters has factors missing in its L-function. (The exact characterization is that the factor for p is missing iff p divides the modulus but not the conductor, but I don’t think it’s necessary to spell this out, it’s kind of obvious from the Euler factor expansion itself.) Also, characters—primitive or imprimitive—“correspond” to L-functions, if anything, not to missing factors, making the new wording clumsy. I would go with the original wording, maybe slightly reformulated so as not to make characters causative agents: Imprimitivity of characters can lead to missing Euler factors in their L-functions.—Emil J. 12:10, 3 September 2012 (UTC)

Looks OK to me. Deltahedron (talk) 21:17, 3 September 2012 (UTC)

Currently there are two statements defining a primitive character: "A character is primitive if it is not induced by any character of smaller modulus" and "A character is primitive if there is no smaller induced modulus". But these aren't the same, the latter is the now common definition of a primitive character, but the former is then incorrect as a character can have smaller induced moduli without being induced by another character. For example, the unique non-trivial character modulo 6 has induced modulus 3 but is not an extension of any character mod 1,2, or 3. Do people agree with me here? — Preceding unsigned comment added by Cjs cjs cjs (talk • contribs) 17:59, 16 May 2013 (UTC)

I don’t get your point. The nontrivial character mod 6 is induced by the nontrivial character mod 3, so it is imprimitive under either definition. If d is an induced modulus of χ, there is indeed a character ψ mod d such that χ is induced by ψ, so the two definitions are equivalent (they are pretty much just saying the same thing in slightly different words).—Emil J. 18:19, 16 May 2013 (UTC)

I was confusing something with induced and extending characters. My bad.

Simple explanation

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 $\omega =e^{2ik\pi /(p-1)}$
• 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 $Y(k)=\sum _{n\in G}\chi (n)e^{2i\pi nk/p}$ the discrete fourier transform of $\chi$ we get that $\forall a\in G,\quad \sum _{n\in G}\chi (an)e^{2i\pi ank/p}=\chi (a)Y(k)$ so that $Y(k)=Y(1){\bar {\chi }}(n)$. from that we can write a Fourier series representation of $\sum _{n}\chi (n)\delta (x-n)={\frac {1}{q}}\sum _{k=1}^{\infty }{\bar {\chi }}(k)\left(Y(1)e^{-2i\pi kx/q}+Y(-1)e^{2i\pi kx/q}\right)$ which leads to the functional equation $L(\chi ,s)=\sum _{n=1}^{\infty }\chi (n)n^{-s}=\int _{0}^{\infty }\sum _{n}\chi (n)\delta (x-n)x^{-s}dx=\sum _{k=1}^{\infty }{\bar {\chi }}(k)k^{s-1}{\frac {1}{q}}\int _{0}^{\infty }\left(Y(1)e^{-2i\pi kx/q}+Y(-1)e^{2i\pi kx/q}-2\right)x^{-s}dx=L(1-s,\chi )A(s)$

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)

New character

(This character is a special example of Dirichlet character)

We define $\left({\frac {a}{n}}\right)$ = 0 if a and n is not coprime, for coprime a and n we define:

$\left({\frac {0}{1}}\right)$ = 1.

$\left({\frac {1}{n}}\right)$ = 1, where n is positive integer.

$\left({\frac {a}{p}}\right)$ = $e^{{2\pi i}/(p-1)}$, where p is prime, and a is the smallest positive primitive root of p.

$\left({\frac {a+kn}{n}}\right)$ = $\left({\frac {a}{n}}\right)$, where a and k are integers, n is positive integer. (i.e. $\left({\frac {a}{n}}\right)$ = $\left({\frac {b}{n}}\right)$ if a ≡ b mod n, where a and b are integers, n is positive integer)

$\left({\frac {ab}{n}}\right)$ = $\left({\frac {a}{n}}\right)\left({\frac {b}{n}}\right)$, where a and b are integers, n is positive integer.

$\left({\frac {a^{k}}{n}}\right)$ = $\left({\frac {a}{n}}\right)^{k}$, where a is integer, k is nonnegative integer, n is positive integer.

$\left({\frac {a}{mn}}\right)$ = $\left({\frac {a}{m}}\right)\left({\frac {a}{n}}\right)$, where a is integer, m and n are positive integers, and gcd(m, n) = 1.

$\left({\frac {1}{4}}\right)$ = 1, $\left({\frac {3}{4}}\right)$ = −1.

$\left({\frac {1}{8}}\right)$ = $\left({\frac {7}{8}}\right)$ = 1, $\left({\frac {3}{8}}\right)$ = $\left({\frac {5}{8}}\right)$ = −1.

$\left({\frac {a}{p^{k}}}\right)$ = $e^{{2\pi i}/((p-1)\cdot p^{k-1})}$, where p is odd prime, k is positive integer, a is the smallest integer such that a ≡ (the smallest primitive root mod p) mod p, and a is a primitive root mod p^k. (a is usually the smallest primitive root mod p^k, but not always, the smallest counterexample is p = 40487)

$\left({\frac {1}{16}}\right)$ = $\left({\frac {15}{16}}\right)$ = 1, $\left({\frac {7}{16}}\right)$ = $\left({\frac {9}{16}}\right)$ = −1, $\left({\frac {3}{16}}\right)$ = $\left({\frac {13}{16}}\right)$ = i, $\left({\frac {5}{16}}\right)$ = $\left({\frac {11}{16}}\right)$ = −i.

$\left({\frac {a}{2^{k}}}\right)$ = $\left({\frac {2^{k}-a}{2^{k}}}\right)$, where k ≥ 3 is integer, a is positive integer. (the definition of $\left({\frac {a}{n}}\right)$ where n is divisible by 32 is more complex)

Thus, if a and n are coprime positive integers, then $\left({\frac {a}{n}}\right)^{\lambda (n)}$ = 1, where $\lambda$ is the Carmichael lambda function (i.e. if a and n are coprime positive integers, then $\left({\frac {a}{n}}\right)$ is a $\lambda (n)$-th root of 1). Besides, $\lambda (n)$ is the smallest positive integer k such that $\left({\frac {a}{n}}\right)^{k}$ = 1 for all positive integers a coprime to n.

The values of $\left({\frac {a}{n}}\right)$ for gcd(a, n) = 1 are (we write these values in $[r,\theta ]$ = $re^{i\theta }$ = $r(\cos \theta +i\sin \theta )$, and since this $r$ is always 1, we only write the $\theta$ of these numbers (in degrees)):

n \ a	comment	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	the trivial character (we define $\left({\frac {0}{1}}\right)$ = 1)	0
2	smallest primitive root mod 2 is 1, thus we define $\left({\frac {1}{2}}\right)$ = $e^{{2\pi i}/(2-1)}$		0
3	smallest primitive root mod 3 is 2, thus we define $\left({\frac {2}{3}}\right)$ = $e^{{2\pi i}/(3-1)}$		0	180
4	special definition for n = 4 (4 = 22, the smallest primitive root mod 2 is 1, and 3 is the smallest a such that a ≡ 1 (mod 2) which is also a primitive root mod 4, thus we define $\left({\frac {3}{4}}\right)$ = $e^{{2\pi i}/((2-1)\cdot 2^{2-1})}$)		0		180
5	smallest primitive root mod 5 is 2, thus we define $\left({\frac {2}{5}}\right)$ = $e^{{2\pi i}/(5-1)}$		0	90	270	180
6	$\left({\frac {a}{6}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{3}}\right)$		0				180
7	smallest primitive root mod 7 is 3, thus we define $\left({\frac {3}{7}}\right)$ = $e^{{2\pi i}/(7-1)}$		0	120	60	240	300	180
8	special definition for n = 8 (the same as the kronecker symbol $\left({\frac {a}{8}}\right)$)		0		180		180		0
9	9 = 32, the smallest primitive root mod 3 is 2, and 2 is the smallest a such that a ≡ 2 (mod 3) which is also a primitive root mod 9, thus we define $\left({\frac {2}{9}}\right)$ = $e^{{2\pi i}/((3-1)\cdot 3^{2-1})}$		0	60		120	300		240	180
10	$\left({\frac {a}{10}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{5}}\right)$		0		270				90		180
11	smallest primitive root mod 11 is 2, thus we define $\left({\frac {2}{11}}\right)$ = $e^{{2\pi i}/(11-1)}$		0	36	288	72	144	324	252	108	216	180
12	$\left({\frac {a}{12}}\right)$ = $\left({\frac {a}{3}}\right)\left({\frac {a}{4}}\right)$		0				180		180				0
13	smallest primitive root mod 13 is 2, thus we define $\left({\frac {2}{13}}\right)$ = $e^{{2\pi i}/(13-1)}$		0	30	120	60	270	150	330	90	240	300	210	180
14	$\left({\frac {a}{14}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{7}}\right)$		0		60		300				120		240		180
15	$\left({\frac {a}{15}}\right)$ = $\left({\frac {a}{3}}\right)\left({\frac {a}{5}}\right)$		0	270		180			90	90			180		270	0
16	special definition for n = 16		0		90		270		180		180		270		90		0
17	smallest primitive root mod 17 is 3, thus we define $\left({\frac {3}{17}}\right)$ = $e^{{2\pi i}/(17-1)}$
18	$\left({\frac {a}{18}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{9}}\right)$
19	smallest primitive root mod 19 is 2, thus we define $\left({\frac {2}{19}}\right)$ = $e^{{2\pi i}/(19-1)}$
20	$\left({\frac {a}{20}}\right)$ = $\left({\frac {a}{4}}\right)\left({\frac {a}{5}}\right)$
21	$\left({\frac {a}{21}}\right)$ = $\left({\frac {a}{3}}\right)\left({\frac {a}{7}}\right)$
22	$\left({\frac {a}{22}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{11}}\right)$
23	smallest primitive root mod 23 is 5, thus we define $\left({\frac {5}{23}}\right)$ = $e^{{2\pi i}/(23-1)}$
24	$\left({\frac {a}{24}}\right)$ = $\left({\frac {a}{3}}\right)\left({\frac {a}{8}}\right)$
25	25 = 52, the smallest primitive root mod 5 is 2, and 2 is the smallest a such that a ≡ 2 (mod 5) which is also a primitive root mod 25, thus we define $\left({\frac {2}{25}}\right)$ = $e^{{2\pi i}/((5-1)\cdot 5^{2-1})}$
26	$\left({\frac {a}{26}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{13}}\right)$
27	27 = 33, the smallest primitive root mod 3 is 2, and 2 is the smallest a such that a ≡ 2 (mod 3) which is also a primitive root mod 27, thus we define $\left({\frac {2}{27}}\right)$ = $e^{{2\pi i}/((3-1)\cdot 3^{3-1})}$
28	$\left({\frac {a}{28}}\right)$ = $\left({\frac {a}{4}}\right)\left({\frac {a}{7}}\right)$
29	smallest primitive root mod 29 is 2, thus we define $\left({\frac {2}{29}}\right)$ = $e^{{2\pi i}/(29-1)}$
30	$\left({\frac {a}{30}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{15}}\right)$ = $\left({\frac {a}{3}}\right)\left({\frac {a}{10}}\right)$ = $\left({\frac {a}{5}}\right)\left({\frac {a}{6}}\right)$ = $\left({\frac {a}{2}}\right)\left({\frac {a}{3}}\right)\left({\frac {a}{5}}\right)$
31	smallest primitive root mod 31 is 3, thus we define $\left({\frac {3}{31}}\right)$ = $e^{{2\pi i}/(31-1)}$
32	special definition for n = 32

Extension

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)

Proposed Rewrite

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)

Rewritten

Done.

"Be bold!"

James in dc (talk) 21:11, 30 June 2021 (UTC)

Vandalism

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)

Undo Vandalism

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)