The Weil–Petersson random prime geodesic conjecture

Let $\mathcal{M}_g$ be the moduli space of closed hyperbolic surfaces of genus $g$, with Weil–Petersson probability measure. For $t>2$, let $\pi_{X_g}(t)$ count oriented primitive closed geodesics of length at most $\ln t$.

Weil–Petersson random prime geodesic conjecture. Theorem 1.1 still holds after replacing the exponent $3/4$ by $1/2$; equivalently, there is a universal constant $c>0$ such that for every $\epsilon>0$ and all admissible functions $t=t(g)$, the probability that

$$\left|\pi_{X_g}(t)-\operatorname{Li}(t)\right|\leq c\\,g\\,t^{1/2+\epsilon}$$

tends to $1$ as $g\to\infty$.

The conjecture is formulated as the random-surface analogue of the Riemann hypothesis. The paper proves the corresponding statement with exponent $3/4+\epsilon$, while the improvement to $1/2$ remains open in the supplied text.