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The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.


I have by now convinced myself that the following is a reasonable conceptualization of why the Riemann zeta function for real $s > 1$ is a natural object of study. First, the probability distributions $\mathbb{P}(X = n) = \frac{1}{\zeta(s)} \left( \frac{1}{n^s} \right)$ on $\mathbb{N}$ are the unique ones satisfying the following two conditions:

  • Given that $n | X$, the probability distribution on $\frac{X}{n}$ is the same as the original distribution, and
  • $\mathbb{P}(X = n)$ is monotonically decreasing.

Second, if one looks at the sequence of measures $\mu_s(A) = \sum_{a \in A} \mathbb{P}(X = a)$ of a subset $A \subset \mathbb{N}$ with respect to the above distribution, then the $s \to 1^{+}$ limit is the logarithmic density of $A$, which agrees with the natural density of $A$ if it exists.

One can also use statistical-mechanical language to describe the above distribution. There is a statistical-mechanical system, the Riemann gas, whose states are the positive integers $n$ and whose energies are the numbers $\log n$, and $\zeta(s)$ is its partition function (which then determines a distribution on $\mathbb{N}$ in the usual way). This explanation conceptualizes, among other things, the von Mangoldt function, whose Dirichlet series is just the average energy of the above system.

However, the language of probability distributions is insufficient for talking about $\zeta(s)$ for $s \le 1$ or for complex $s$.


Is there a way to conceptualize the values of the zeta function at complex values of $s$ as a "Wick rotation" of its values at real $s$? That is, is there some reasonable quantum-mechanical interpretation of numbers like the "formal" measure

$$\mu_{s+it}(A) = \frac{1}{\zeta(s+it)} \sum_{a \in A} e^{-(s + it)\log a}$$

(for $s, t$ real) as a probability amplitude, or something along those lines? Does this reasonable quantum-mechanical interpretation single out the critical line $s = \frac{1}{2}$?

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I found a paper that mentions both Wick rotations and zeta functions. I have no idea whether it does what you want. J Casahorran, Quantum-mechanical tunneling: differential operators, zeta-functions and determinants, Fortschr. Phys. 50 (2002), no. 3-4, 405–424, MR 2003k:81061. –  Gerry Myerson Nov 17 '10 at 2:48
Thanks, Gerry. Unfortunately reading the abstract it sounds like the paper is using zeta-function regularization and not directly talking about zeta functions. –  Qiaochu Yuan Nov 17 '10 at 9:25
The Casahorran paper is available for free on arXiv.org: arxiv.org/pdf/quant-ph/0011059 –  mathphysicist Aug 22 '11 at 15:32
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1 Answer

The Riemann zeta function $\zeta(s)$ at complex $s$ has the statistical physics interpretation of a partition function at complex temperature. This has no direct physical meaning in general, but for certain models it does. A notable example is the Ising model, where the real and imaginary temperature axes are related by a transformation from an hexagonal to a triangular lattice.

Quite generally, the zeroes of the partition function in the complex plane fall on lines rather than in areas. For ferromagnetic models this is the content of the Yang-Lee theorem. It is therefore natural to expect the Riemann hypothesis to hold, although the Yang-Lee theorem does not cover this case.

An overview of the older literature on complex temperature partition functions is: "Location of zeros in the complex temperature plane: Absence of Lee-Yang theorem", W. van Saarloos and D. A Kurtze, J. Phys. A: Math. Gen. 17 (1984) 1301-1311. A more recent paper is "Complex-temperature partition function zeros of the Potts model on the honeycomb and kagome ́ lattices", H. Feldmann, R. Shrock, and S.-H. Tsai, Phys. Rev. E 57, 1335 (1998). There are many more papers, it is a quite active field of study.

A very recent paper is http://arxiv.org/pdf/1110.0942

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