Open Circular Billiards and the Riemann Hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g., a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeros of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.

Figure 1

Geometry of the billiard.

Figure 2

Poles of $\tilde{P}(s)$ [Eq. (9)] for $q=1$ (one hole) at $-2$, odd integers less than or equal to 1, and nontrivial values with real part $-1/2$ assuming the Riemann hypothesis. The contour of Eq. (5) is given.

Figure 3

The scaled survival probability $ϵtP(t)$ [see Eq. (4)] appears to approach a limiting function as $ϵ\to 0$ with $ϵt$ held constant. Here we use ${10}^8$ random initial conditions, $\theta =1$ (an irrational multiple of $\pi$) and the curves are $ϵ={10}^{-n/2}$ with $n=0\cdots 6$. At large $ϵt$ the function converges to unity, consistent with Eq. (13).

Figure 4

Numerical computation of $P_{\infty }^{\prime \prime }(ϵ)$ for the single hole case $q=1$, using Eq. (9) and varying the number of poles considered on the critical line; real poles are considered for $s>-10$ in all plots. From (4) it is seen that the second derivative of $P_{\infty }$ is a function with uniformly spaced steps and constant average gradient in the variables shown on the axes. Convergence is evident except for very small ${ϵ}^{-1}$ for which more real poles would be required; large ${ϵ}^{-1}$ require more critical poles for the steps to be visible.