Talk:Dirichlet L-function

The notation in this article is poor: switching k's and q's, sometimes writing L(X,s) and sometimes writing L(s,X)... — Preceding unsigned comment added by 128.112.87.62 (talk • contribs) 18:15, 16 July 2007 (UTC)

On the article of Dirichlet L function

The text of reason seems to be cut maybe by character limit. So I put the whole text (minor changed) here:

"One source of the usual symbol for the root number is https://encyclopediaofmath.org/wiki/Artin_root_numbers. I can show you many sources (papers) other than this. And f is intended to stand for the conductor (the conductor usually denoted by f or sometimes c, as you know (I hope)).

The biggest problem is the definition of complete L function Λ (the usual symbols are Λ, L^\tilde, L^* and so on. ξ is not included(, this is used for complete Dedekind zeta)). This is absolutely inappropriate, because it should be a special form of the complete Hecke L function (see for example Neukirch "Algebraic Number Theory" p.503) but the previous one is not so. Other symbols are certainly my favourite, so I don't complain if you change those." Ys1123 (talk) 08:18, 20 July 2024 (UTC)

@Ys1123: thanks for the comments. I have no doubt that you are more familiar with algebraic number theory than me, so I appreciate your viewpoint.

Overall, I would say we should not switch to f. The other changes like δ I'm willing to accept.

For modulus vs. conductor, I think Neukirch p. 434–443 is a good illustration: he writes χ mod m and conductor f; but whenever χ has already been assumed to be primitive (as is the case here), he writes m and not f. Generally sources that use q or other letters for the modulus don't switch from that letter to a different letter just because χ is primitive.

You brought up Hecke L functions and such, which are fair and natural connections to make. I would hesitate against giving such matters too much weight, as, generally speaking, there is no requirement that styles of notation and convention must be invariant between a topic and generalizations of the topic. In the balance among sources, the greatest weight is on those that are directly about Dirichlet L functions. For example, https://encyclopediaofmath.org/wiki/Artin_root_numbers is written with some choices of notation, while https://encyclopediaofmath.org/wiki/Dirichlet_L-function is written with different choices. The latter view matters more for purposes of this article. I think there is nothing unusual about ξ, but of course nothing unusual about Λ either.

Davenport, chapter 9	$q$	${\mathfrak {a}}$	$\xi (s,\chi )$	No symbol
Montogomery and Vaughan, p. 333	$q$	$\kappa$	$\xi (s,\chi )$	$\varepsilon (\chi )$
Iwaniec and Kowalski, p. 84	$q$	$\kappa$	$\Lambda (s,\chi )$	$\varepsilon (\chi )$
Apostol, p. 274	$k$	$a$	$\xi (s,\chi )$	$1/\varepsilon (\chi )$
https://encyclopediaofmath.org/wiki/Dirichlet_L-function	$d$	$\delta$	$\xi (s,\chi )$	$1/\varepsilon (\chi )$
Neukirch, p. 440	$m$	$p$	$\Lambda (\chi ,s)$	$W(\chi )$

Adumbrativus (talk) 09:02, 21 July 2024 (UTC)

OK, on the modulus, I can accept your claim (but I don't think it must be changed). So you can change f to another symbol.

I want to remark on the symbol ε, there is confusing concept "ε factor" in the theory of Tate's thesis, it is not equal to the root number (but similar in certain sense). So I think it is not good idea using the symbol ε.

On the complete Dirichlet L, I know Davenport, Montgomery etc write as such way and there are many descriptions relying on them unfortunately.

But I would say nothing should be more mathematically-natural in its descriptions than sources of information that are as well-known and widely viewed as Wikipedia. If it isn't so, inappropriate descriptions increase and make people confused (in fact, I'm one of the people who confused by the description).

Wikipedia is also viewed by many mathematicians who have many backgrounds (as for me, I am studying the Tate's thesis). To put it dramatically, I cannot stand by and watch the world get even a little worse. Ys1123 (talk) 11:16, 21 July 2024 (UTC)

Ok so we now have $q,\delta ,\Lambda ,W$ which appears to be consensus-acceptable based on the discussion.

Separately, which I forgot to mention before: we now have

$\Lambda (s,\chi )=q^{s/2}\pi ^{-(s+\delta )/2}\operatorname {\Gamma } \left({\frac {s+\delta }{2}}\right)L(s,\chi ).$

Do you have some reason to have picked $q^{s/2}\pi ^{-(s+\delta )/2}$? I see some sources with $(q/\pi )^{s/2}$ and others with $(q/\pi )^{(s+\delta )/2}$. Adumbrativus (talk) 06:52, 22 July 2024 (UTC)

@Adumbrativus That's why just it is a special form of complete Hecke L.

OK, I would explain in detail.

The complete Hecke L function is Λ (s, χ)=(|D_F| \mathcal{N}{\mathfrak{f}_{\chi}}}^{s/2} (Gamma factor) L(s, χ)

Now since F= ℚ, the Gamma factor is the one at real place Γ _ ℝ ((s+ δ)/2) where Γ_ ℝ (s, χ)= π ^{-s} Γ (s).

Thus, Λ (s, χ)=f^{s/2} π ^{-((s+ δ)/2)} Γ ((s+ δ)/2) L(s, χ). Ys1123 (talk) 10:59, 22 July 2024 (UTC)

Ok, I see where you're coming from and I'm fine with that. Thanks again! Adumbrativus (talk) 04:47, 23 July 2024 (UTC)