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Archive for November, 2010

Boolean rings, ultrafilters, and Stone’s representation theorem

Posted in math.AC, math.CT, math.GN, math.LO, tagged , , on November 22, 2010| 12 Comments »

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter.

Terence Tao has already written a great introduction to ultrafilters with an eye towards nonstandard analysis. I’d like to introduce them from a different perspective. Some of the topics below are also covered in these posts by Todd Trimble.

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A little more about zeta functions and statistical mechanics

Posted in math.NT, math.PR, physics.stat-mech, tagged , on November 14, 2010| 1 Comment »

In the previous post we described the following result characterizing the zeta distribution.

Theorem: Let a_n = \mathbb{P}(X = n) be a probability distribution on \mathbb{N}. Suppose that the exponents in the prime factorization of n are chosen independently and according to a geometric distribution, and further suppose that a_n is monotonically decreasing. Then a_n = \frac{1}{\zeta(s)} \left( \frac{1}{n^s} \right) for some real s > 1.

I have been thinking about the first condition, and I no longer like it. At least, I don’t like how I arrived at it. Here is a better way to conceptualize it: given that n | X, the probability distribution on \frac{X}{n} should be the same as the original distribution on X. By Bayes’ theorem, this is equivalent to the condition that

\displaystyle \frac{a_{mn}}{a_n + a_{2n} + a_{3n} + ...} = \frac{a_m}{a_1 + a_2 + ...}

which in turn is equivalent to the condition that

\displaystyle \frac{a_{mn}}{a_m} = \frac{a_n + a_{2n} + a_{3n} + ...}{a_1 + a_2 + a_3 + ...}.

(I am adopting the natural assumption that a_n > 0 for all n. No sense in excluding a positive integer from any reasonable probability distribution on \mathbb{N}.) In other words, \frac{a_{mn}}{a_m} is independent of m, from which it follows that a_{mn} = c a_m a_n for some constant c. From here it already follows that a_n is determined by a_p for p prime and that the exponents in the prime factorization are chosen geometrically. And now the condition that a_n is monotonically decreasing gives the zeta distribution as before. So I think we should use the following characterization theorem instead.

Theorem: Let a_n = \mathbb{P}(X = n) be a probability distribution on \mathbb{N}. Suppose that a_{nm} = c a_n a_m for all n, m \ge 1 and some c, and further suppose that a_n is monotonically decreasing. Then a_n = \frac{1}{\zeta(s)} \left( \frac{1}{n^s} \right) for some real s > 1.

More generally, the following situation covers all the examples we have used so far. Let M be a free commutative monoid on generators p_1, p_2, ..., and let \phi : M \to \mathbb{R} be a homomorphism. Let a_m = \mathbb{P}(X = m) be a probability distribution on M. Suppose that a_{nm} = c a_n a_m for all n, m \in M and some c, and further suppose that if \phi(n) \ge \phi(m) then a_n \le a_m. Then a_m = \frac{1}{\zeta_M(s)} e^{-\phi(m) s} for some s such that the zeta function

\displaystyle \zeta_M(s) = \sum_{m \in M} e^{-\phi(m) s}

converges. Moreover, \zeta_M(s) has the Euler product

\displaystyle \zeta_M(s) = \prod_{i=1}^{\infty} \frac{1}{1 - e^{- \phi(p_i) s}}.

Recall that in the statistical-mechanical interpretation, we are looking at a system whose states are finite collections of particles of types p_1, p_2, ... and whose energies are given by \phi(p_i); then the above is just the partition function. In the special case of the zeta function of a Dedekind abstract number ring, M = M_R is the commutative monoid of nonzero ideals of R under multiplication, which is free on the prime ideals by unique factorization, and \phi(I) = \log N(I). In the special case of the dynamical zeta function of an invertible map f : X \to X, M = M_X is the free commutative monoid on orbits of f (equivalently, the invariant submonoid of the free commutative monoid on X), and \phi(P) = \log |P|, where |P| is the number of points in P.

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Zeta functions, statistical mechanics and Haar measure

Posted in math.GR, math.NT, math.PR, physics.stat-mech, tagged , , , , on November 9, 2010| 3 Comments »

An interesting result that demonstrates, among other things, the ubiquity of \pi in mathematics is that the probability that two random positive integers are relatively prime is \frac{6}{\pi^2}. A more revealing way to write this number is \frac{1}{\zeta(2)}, where

\displaystyle \zeta(s) = \sum_{n \ge 1} \frac{1}{n^s}

is the Riemann zeta function. A few weeks ago this result came up on math.SE in the following form: if you are standing at the origin in \mathbb{R}^2 and there is an infinitely thin tree placed at every integer lattice point, then \frac{6}{\pi^2} is the proportion of the lattice points that you can see. In this post I’d like to explain why this “should” be true. This will give me a chance to blog about some material from another math.SE answer of mine which I’ve been meaning to get to, and along the way we’ll reach several other interesting destinations.

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