Has the presumption of the Riemann Hypothesis had any impact on probability? ie: Are there any important theorems in probability that begin with "Give that the RH is true..." (It seems likely, given the apparently random walk of $\pi(x)$, similar "seeds" are seen in many real life instances.)
2026-03-25 11:08:35.1774436915
Probability and discrete mathematics
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The Riemann hypothesis is connected to random matrix theory, which also plays a role for statistics and physics (e.g., quantum chaos, energy levels). However, a theorem starting with "Assume that the Riemann hypothesis holds" usually appears in number theory. My impression is that the important consequences of the Riemann hypothesis concern number theory, and not probability theory. The Riemann hypothesis has a natural generalisation, namely the Grand Riemann Hypothesis (GRH), which would imply many important results in various branches of number theory. See the survey paper of Peter Sarnak.
On the other hand, methods from probability theory can be useful for the Riemann zeta function and prime number theory, i.e., for the distribution of primes.