# Introduction to L-series and Dirichlet characters?

I'm looking for an introductory text on Dirichlet characters and the L-series of a field K, specifically for quartic extensions of $\mathbb{Q}$. I have Davenport's Multiplicative Number Theory, Ireland/Rosen, Marcus' Number Fields, and Washington's Cyclotomic Fields, but things seem to be scattered and I was hoping for more concrete examples. Does anyone have any suggestions? This graduate student would appreciate any ideas you have!

Thanks!

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Can you be more specific about what kind of examples you are looking for? –  Álvaro Lozano-Robledo Jul 21 '11 at 23:21
I can't imagine there is a text with an extended discussion of L-functions of quartic fields. I also don't understand the impression of things being "scattered". There are one or two chapters in Ireland and Rosen on Dirichlet L-functions; it's not smeared all around the book. Try looking in Knapp's book on elliptic curves for another treatment of Dirichlet L-functions. To a certain extent you need to work out examples yourself too. –  KConrad Jul 21 '11 at 23:59
Chapter 7 of Neukirch, Algebraic Number Theory, has a nice treatment of L-functions of number fields. I think your best bet is to read Davenport first, and then (assuming you know some algebraic number theory already) read Neukirch. You might also read Tate's thesis for a beautiful alternative treatment; it makes reasonably beginner-friendly reading. I also second KConrad's advice that you should write down a ton of examples yourself. –  Frank Thorne Jul 22 '11 at 0:45
At a truly introductory level, you may enjoy reading through Apostol's "Introduction to Analytic Number Theory". Chapter 6 deals with Dirichlet characters and Chapters 11 and 12 develop the theory of Dirichlet series. But for a treatment of Dedekind zeta functions you will have to go elsewhere. –  Álvaro Lozano-Robledo Jul 22 '11 at 0:52
Thank you for all of your suggestions! I'm going to check out both Apostol and Neukirch, and Frank's suggestion to start with Davenport was really helpful. –  MathHands Jul 24 '11 at 18:26

As suggested by others, the books by Apostol, Marcus, Washington, Neukirch, Frohlich and Taylor, should do a good job covering the theory, and some examples. But if you want explicit examples, you will probably have to work those out yourself (as pointed out by KConrad). Use Sage!

The functions specific to Number Fields are listed here:

Look also here for extended functionality:

In particular, if K is a number field, K.zeta_function() is the Dedekind zeta function of K and K.zeta_coefficients(n) returns the first n coefficients of the Dedekind zeta function of this field as a Dirichlet series.

Here is an example of code to compute values of the Dedekind zeta function for a biquadratic field:

P.<x>=PolynomialRing(Integers());

K.<k>=NumberField([x^2+1,x^2-5]);

F.<f> = K.absolute_field();

Z=F.zeta_function();

Then

Z(1.0000000000001) returns 4.74937139529845e12, since there is a simple pole at $s=1$, and F.zeta_coefficients(100) returns

[1, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3]

You can also try Z.taylor_series(a,k) to obtain the first k coefficients of the Taylor series for Z(z) around z=a. In this case, Z.taylor_series(2,10) returns

1.20029545506816 - 0.392371605671893*z + 0.466197993407214*z^2 - 0.476088948200672*z^3 + 0.474236081878810*z^4 - 0.473954466352746*z^5 + 0.474527990929374*z^6 - 0.474871501779465*z^7 + 0.474954910327522*z^8 - 0.474952153099398*z^9 + O(z^10)

Edit to add: You may also want to check that the analytic class number formula works:

RR = Reals();

2^(F.signature()[0])*(2*RR(pi))^(F.signature()[1])*F.class_number()*F.regulator()/(len(F.roots_of_unity())*sqrt(F.discriminant()))

returns 0.474937034646450

and Z(1.00000000000001)*(1.00000000000001 - 1) = 0.474935594750836 ... close enough!

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This is a very good suggestion. Wish I thought of it! –  Daniel Larsson Jul 22 '11 at 13:59

The last chapter of Fröhlich and Taylor's "Algebraic number theory" covers Dirichlet L-functions. The last section of that chapter contains applications to biquadratic fields (as well as cubic and sextic fields). There are also some "purely algebraic" results on biquadratic fields earlier in the book.

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Nancy Childress' book on class field theory has a nice chapter, even though it is more or less taken (in parts) from Washington's chapter 3 and 4.

Lang's "Algebraic number theory" also has a chapter I think.

Serre's "A course in arithmetic" is also nice.

Edit: there is a rather thorough introduction to L-functions in Volume 2 of Cohen's Number theory. I haven't seen it myself, but from the table of contents it seems very detailed and from scratch-approach.

As Kevin mentions Fröhlich-Taylor has a chapter on L-functions, but I find their book a little hard to read. But you might feel differently.

I'll report back if I think of anything else.

Cheers,

/Daniel

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