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.

### Background

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 $s$.

### Question

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 $s = \frac{1}{2}$?

This post imported from StackExchange MathOverflow at 2015-01-08 14:06 (UTC), posted by SE-user Qiaochu Yuan