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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Newton’s sums, necklace congruences, and zeta functions II

In a previous post I gave essentially the following definition: given a discrete dynamical system, i.e. a space and a function , and under the assumption that has a finite number of fixed points for all , we define the dynamical zeta function to be the formal power series . What I didn’t do was [...]

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The cyclotomic identity and Lyndon words

In number theory there is a certain philosophy that is a good toy model for the integers . The two rings share an important property: they are basically the canonical examples of Euclidean domains, hence PIDs, hence UFDs. However, many number-theoretic questions involving prime factorization over are much easier than their corresponding questions over . [...]

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Newton’s sums, necklace congruences, and zeta functions

The goal of this post is to give a purely combinatorial proof of Newton’s sums which would have interrupted the flow of the previous post. Recall that, in the notation of the previous post, Newton’s sums (also known as the first Newton-Girard identity) state that . One way to motivate a combinatorial proof is to [...]

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