TITLE: Consequences of the Riemann hypothesis
QUESTION [146 upvotes]: I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice to include consequences of the generalized Riemann hypothesis (but specify which one is assumed).

REPLY [3 votes]: 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