Posted by: holdenlee | June 27, 2014

Analytic Class Number Formula


\displaystyle \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots =\sum_{n=0}^{\infty}\frac{1}{4n+1}-\frac{1}{4n+3}.

I proved this identity in a post several years ago on Number Theory and Pi. It turns out that this is a special case of a more general identity called the analytic class number formula (ACNF) that evaluates an analytic quantity–a L-function at 1, L(1,\chi)–in terms of algebraic information about an extension of \mathbb Q. I have just added a chapter on the ACNF in the open-source number theory text (currently chapter 35 in the text) (source: in analytic-chapters/chapter4.tex). (Thinking in terms of L-series makes the previous proof less mysterious: fact that the number of ways to write a number as a sum of 2 squares is 4(d_1(n)-d_3(n)) falls out of an identity of Dirichlet series along with understanding how sum-of-squares relates to splitting in \mathbb Q(i).)

The ACNF also gives an identity which relates a class number to the distribution of quadratic residues modulo a prime p. In the post on prime-producing polynomials I noted that when the class number is 1, more quadratic nonresidues clump near the beginning. The ACNF gives a precise statement of this fact: for p\equiv 3\pmod 4,

h_{\mathbb Q(\sqrt{-p})}=\frac{1}{2-\left(\frac{2}{p}\right)}(R-N)

where R,N are the number of quadratic residues and nonresidues in (0,\frac p2), respectively. In other words, there are more quadratic than nonquadratic residues near the beginning, with the difference more pronounced when the class number is higher. When the class number is 1 this difference is “as small as can be.”

Some (meta-)notes on writing:

I tried to come up with as much of the proof of the ACNF on my own as I could, so this took an unusually long time to write. I glanced at PROMYS notes–which goes slowly through the ACNF for various quadratic fields (my “thread” for quadratic fields follows his)–then tried to adapt it to the general case, only referring to Borevich and Shafarevich when needed. As a result I think the “change of variables” approach I used is cleaner.

Many people have said that

  1. the best way to learn a math theorem is to try to prove it on your own, but that
  2. this takes a LOT of time.

(I can’t find an exact reference for this right now, but it’s been talked about numerous times on math.stackexchange.) I believe the learning process can be sped up if we write math texts that are geared towards helping the student “come up” with the proof (“guided exploration”). (Many math texts instead tend to encourage line-by-line verification rather than guessing what the next step is. I’m interested in recommendations for books that follow the guided exploration model, so please leave a comment!)

Hence I wrote the chapter in a style which I want to standardize for the number theory text , and hope will be adopted by other mathematical writers. I

  1. start with some mathematical “gem” that captures a reader’s interest, (here it’s Ramanujan-style, i.e., without explanation to get the reader to think about it; here I copy Mazur in his article in PCTM article on algebraic number theory),
  2. state the main result and give a thread for the reader to follow to prove a “baby” version of it that captures the main ideas–here the baby version is the case for quadratic fields, and the questions are presented in an order so that the reader has to take into account a new “obstacle” every time: for example, what happens when the class number is not 1? Then write the solutions. (Learning a baby version completely helps very much; see “slowing down to speed up.” Also see “proofs as obstacle avoidance.”)
  3. give a complete proof for the general case. (Someone who learns very quickly/abstractly can just read this, if (s)he likes.) I try to parallel the baby case and delimit where the more technical parts are. The aim isn’t necessarily to have the reader come up with the whole proof–it’s easy to get mired in technical details–but instead to really understand the skeleton so that just scanning the proof, you can see how the technical parts fit into that skeleton.

In summary, while writing a piece of math, many people tend to write it with a “verification” mindset–making sure the lemmas, etc. follow soundly one after another. The writer (lecturer) would learn the material well; however not so much the reader (student), because the decisions about what the next step is, and how to connect the various parts, are going on in the writer’s mind and not made known to the student. Rather than just communicating (writing, lecturing) math as a series of “verifications,” we can communicate it as a productive thread of questions that accelerate a student on the path to finding a proof.

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Please leave comments on the article either below or on the number theory googledoc.

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