Update

I put up a post over at the StackOverflow blog describing a little of what I’ve been up to this summer. Curiously enough, the Zipf distribution which shows up in that post is the same as the zeta distribution that shows up when trying to motivate the definition of the Riemann zeta function. I’m sure [...]

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A little more about zeta functions and statistical mechanics

In the previous post we described the following result characterizing the zeta distribution. Theorem: Let be a probability distribution on . Suppose that the exponents in the prime factorization of are chosen independently and according to a geometric distribution, and further suppose that is monotonically decreasing. Then for some real . I have been thinking [...]

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Zeta functions, statistical mechanics and Haar measure

An interesting result that demonstrates, among other things, the ubiquity of in mathematics is that the probability that two random positive integers are relatively prime is . A more revealing way to write this number is , where is the Riemann zeta function. A few weeks ago this result came up on math.SE in the [...]

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Walks on graphs and statistical mechanics

I finally learned the solution to a little puzzle that’s been bothering me for awhile. The setup of the puzzle is as follows. Let be a weighted undirected graph, e.g. to each edge is associated a non-negative real number , and let be the corresponding weighted adjacency matrix. If is stochastic, one can interpret the [...]

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