Wick rotation and the Riemann zeta function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:30:04Z http://mathoverflow.net/feeds/question/46008 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46008/wick-rotation-and-the-riemann-zeta-function Wick rotation and the Riemann zeta function Qiaochu Yuan 2010-11-14T03:00:56Z 2011-10-06T20:27:01Z <p>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.</p> <h3>Background</h3> <p>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:</p> <ul> <li>Given that $n | X$, the probability distribution on $\frac{X}{n}$ is the same as the original distribution, and</li> <li>$\mathbb{P}(X = n)$ is monotonically decreasing.</li> </ul> <p>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. </p> <p>One can also use statistical-mechanical language to describe the above distribution. There is a statistical-mechanical system, the <a href="http://en.wikipedia.org/wiki/Primon_gas" rel="nofollow">Riemann gas</a>, 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 <a href="http://en.wikipedia.org/wiki/Von_Mangoldt_function" rel="nofollow">von Mangoldt function</a>, whose Dirichlet series is just the average energy of the above system.</p> <p>However, the language of probability distributions is insufficient for talking about $\zeta(s)$ for $s \le 1$ or for complex $s$. </p> <h3>Question</h3> <p>Is there a way to conceptualize the values of the zeta function at complex values of $s$ as a <a href="http://en.wikipedia.org/wiki/Wick_rotation#Statistical_and_quantum_mechanics" rel="nofollow">"Wick rotation"</a> of its values at real $s$? That is, is there some reasonable quantum-mechanical interpretation of numbers like the "formal" measure </p> <p>$$\mu_{s+it}(A) = \frac{1}{\zeta(s+it)} \sum_{a \in A} e^{-(s + it)\log a}$$</p> <p>(for $s, t$ real) as a probability amplitude, or something along those lines? Does this reasonable quantum-mechanical interpretation single out the critical line $s = \frac{1}{2}$? </p> http://mathoverflow.net/questions/46008/wick-rotation-and-the-riemann-zeta-function/73409#73409 Answer by Carlo Beenakker for Wick rotation and the Riemann zeta function Carlo Beenakker 2011-08-22T14:59:37Z 2011-10-06T20:27:01Z <p>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. </p> <p>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.</p> <p>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.</p> <p>A very recent paper is <a href="http://arxiv.org/pdf/1110.0942" rel="nofollow">http://arxiv.org/pdf/1110.0942</a></p>