algebraic-number-theorycharactersdirichlet-seriesnt.number-theorytextbook-recommendation
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!
You can find the answer in the history of the subject. For brevity let us consider the following two genuinely number theoretic questions that were of great interest already to Gauss (and Fermat, Euler, Lagrange, Legendre, Jacobi, Dirichlet, Eisenstein):
(1) For which primes is a given integer a quadratic residue?
(2) Which numbers can be written as a sum of three squares and in how many ways?
These questions were pretty well understood by Gauss and his contemporaries, but they admit equally natural generalizations which turned out to be much much harder (and they are being studied until the present day):
(3) Given an irreducible polynomial over the integers, over which primes decomposes the polynomial in a particular way (e.g. splits into linear factors)?
(4) Which numbers are represented by a positive integral ternary quadratic form and in how many ways? How do the representations distribute on the corresponding ellipsoid?
The best answers to these questions rely heavily on the theory of automorphic forms and their $L$-functions. Question (3) leads naturally to Artin $L$-functions, the question if there is an alternate way to describe their coefficients, for which the best answers are produced by the Langlands program. Question (4) leads naturally to the theory of genera and spinor genera, theta series and cusp forms, Siegel's maass formula and Eisenstein series, Siegel's bound for the class number, the Shimura lift and Waldspurger's formula, bounds for automorphic $L$-functions, all which necessitate a global automorphic thinking. Question (4) also leads to more general questions such as representing a quadratic form by another one, or their counterparts over number fields, which brings to surface a wider class of automorphic forms (e.g. Siegel and Hilbert modular forms).
Automorphic forms and their $L$-functions is not a digression from the natural numbers but the most fitted tool to formulate and study their properties. I often wonder which comes first: the natural numbers, or automorphic forms?
I understand I have not answered your question completely. My aim was to indicate two hot spots (Galois representations, and quadratic forms) where automorphic forms have had a tremendous influence, including contemporary research.
I will go out on a limb and say that in my opinion, it is the norm, rather than the exception, for a branch of mathematics to be a collection of results that we can prove using the techniques we know, rather than a beautiful coherent theory. As a student, one is exposed to a biased sample—the areas of mathematics that have been sufficiently developed to form a systematic theory are precisely the areas that lend themselves to the "coherent development" that you mentioned. This can create an illusion that all areas of mathematics are like that.
If humanity were collectively much more intelligent than it is, our textbook on analytic number theory would probably first prove "Theorem 1: The Generalized Riemann Hypothesis" and then go on to prove all the results that are predicted by pretending that the prime numbers are random but that don't immediately follow from GRH. This would be a beautifully coherent treatment of the subject. But since we're so woefully far away from being able to prove such things, such a textbook is simply not an option.
Having said that, I do agree with you that some textbooks could present more forest and fewer trees than they do. One text I'd recommend is Donald Newman's Analytic Number Theory. Newman rightly emphasizes that the first thing you have to get thoroughly comfortable with is the idea that you can study numbers by studying their generating functions. Newman goes through several examples, starting with easy ones and working up to harder ones. Getting completely comfortable with generating functions is so important that I'd even recommend that you spend some time with books such as Wilf's generatingfunctionology or Flajolet and Sedgewick's Analytic Combinatorics just to get a feeling for what generating functions can do for you. Wilf and Flajolet–Sedgewick are interested in combinatorics rather than number theory and so focus on ordinary generating functions rather than Dirichlet series, but any increase in your comfort level with generating functions will pay dividends in your study of analytic number theory. You will probably recall that Apostol's book spends a lot of time studying different generating functions and formulating many different equivalent statements of (for example) the prime number theorem. I found all these manipulations mysterious and unmotivated the first time I encountered them, but once you understand the value of packaging information in a generating function in different ways, the manipulations will seem less baffling.
The next big idea in analytic number theory is that complex analysis, in particular the study of zeros and singularities, gives you asymptotic information about generating functions. Again, the Flajolet–Sedgewick book gives many examples of this general principle for ordinary (and exponential) generating functions, where it's easier to get a feel for why there is a connection between singularities in the complex plane and asymptotics. For analytic number theory one must also study Dirichlet series, where there are more technical difficulties, but at a high level, the basic idea is the same. I like the treatment in Serre's Course in Arithmetic, which proves in a unified way some basic properties that hold for both Dirichlet series and ordinary power series, and makes it clear where the analogy between the two starts to break down. (Also Serre's treatment of Dirichlet's theorem on arithmetic progressions is very clean; historically, Dirichlet's theorem was arguably the first big result of analytic number theory, so it's a good theorem to master thoroughly at an early stage of your study.) The treatment of Dirichlet series in Montgomery and Vaughan's Multiplicative Number Theory is also excellent.
At some point I think it is worth studying Riemann's paper on the zeta function. The book by Harold Edwards Riemann's Zeta Function provides an excellent guided tour of Riemann's paper. John Derbyshire's book Prime Obsession is also worth looking it; it's semi-popular but gives enough technical details for you to understand what Riemann's exact formula actually says. I think that a thorough understanding of Riemann's exact formula will be a big step forward in getting the bird's-eye view of analytic number theory that you're seeking.
Finally, for a short overview of the whole subject, I think that Andrew Granville's article in the Princeton Companion to Mathematics is hard to beat. It goes into subjects such as sieve theory and prime gaps that I haven't mentioned here.
Best Answer
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!
You can download Sage for free at www.sagemath.org
The functions specific to Number Fields are listed here:
http://www.sagemath.org/doc/reference/sage/rings/number_field/number_field.html
Look also here for extended functionality:
http://www.sagemath.org/doc/reference/lfunctions.html
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!