I haven't seen this application mentioned:

The closed horocycle $\gamma_l$ of length $l$ of the modular surface equidistributes to the hyperbolic volume form $\omega$ when $l\to +\infty$, and RH is equivalent to the error term:
$$\frac{1}{l} \int_{\gamma_l} f = \int f \omega  + o(l^{-3/4+\epsilon})$$
for every smooth function with compact support.

This is due to Don Zagier. https://people.mpim-bonn.mpg.de/zagier/files/scanned/EisensteinRiemannZeta/eisenstein-zeta-978-3-662-00734-1_10.pdf

Verjovsky has another reformulation of the Riemann hypothesis in terms of convergence of measures. https://projecteuclid.org/journals/kodai-mathematical-journal/volume-17/issue-3/Discrete-measures-and-the-Riemann-hypothesis/10.2996/kmj/1138040054.full