Applications of Riemann Hypothesis outside number theory

Question

I'm trying to write a survey article about Riemann Hypothesis, especially about its corollaries and analogies in other fields. I found that there are tons of results in number theory (especially about prime numbers) that can be proved by assuming RH. Also, there's an interesting story about Stark-Heegner theorem related to RH. However, it is hard to find its application in other fields. Are there any interesting corollaries that follows from RH, but not in number theory? (Not even in mathematics? Maybe Physics?) Thanks in advance.

Answer 1

Apart from other areas in math other than number theory, it seems the main field with connections to the Riemann Hypothesis is physics. For example, there's the summary of Marek Wolf's $1999$ preprint in Applications of statistical mechanics in prime number theory (Budapest lecture notes) which states

The functional equation allows the definition to be extended to the other half of the complex plane. It turns out that this equation is analogous to the Kramers–Wannier duality relation [$8$] for the partition function of the $2$-dimensional Ising model. This suggests that the zeta function is in some sense acting as a partition function, which is the primary quantity in statistical mechanics. Two papers are mentioned which examine this analogy. In [$9$] Bernard Julia defines an abstract numerical "gas" whose partition function is the zeta function. In [$10$] A. Knauf finds a spin system whose partition function is the ratio

$$Z(\beta) = \zeta(\beta - 1)/\zeta(\beta)$$

Lee–Yang [$11$] type theorems can be used to localise the zeros of partition functions, and the Riemann Hypothesis concerns the restriction of the zeros of the zeta function. Hence an interpretation of the zeta function as a partition function opens up the possibility of applying Lee–Yang theorems to the problem of proving the RH.

It then goes on to discuss another approach to proving the RH using spectral interpretation. Later, it also says

In the previously mentioned article [$9$], B.L. Julia constructs the "free Riemann gas", an abstract thermodynamic system based on the prime numbers. It's a fairly uncontrived notion, with the primes playing the role of the "atomic" particles, each $p_n$ having energy $\log p_n$. Most importantly, its partition function is the Riemann zeta function. Wolf has constructed a variant on this – his "prime gas" also has the primes acting as particles, but in this case $p_n$ has energy $p_{n+1} – p_n$, i.e. the distance to the next prime.

Justina R. Yang's paper The Riemann Hypothesis: Probability, Physics and Primes has in its Introduction on page $1$

The Riemann Hypothesis is also related to applied math and science — particularly fields such as statistics and physics. Because of this, ideas stemming from the fields of probability theory or the study of subatomic particles could very well be the key to solving the Riemann Hypothesis — and, by extension, to the multitudes of other math problems that are similar to the Riemann Hypothesis.

Later, in the "The Zeta Function’s Zeros and Physics" section starting on page $27$, it explains

Then, in $1972$, the American number theorist Hugh Montgomery made a discovery that not only supported the Hilbert-Pólya Conjecture but also linked the Riemann zeta function and Hypothesis to physics. Montgomery had been studying the zeros of the zeta function — specifically, the distances between those consecutive, non-trivial zeros that lie on the critical line. (Note that these distances correspond exactly to those between the real values of such that $\xi(t) = 0$.) Montgomery eventually discovered an expression that represented the statistical distribution of those distances, and after a chance meeting, told his result to physicist Freeman Dyson, of the Princeton Institute for Advanced Study. Dyson immediately said that Montgomery’s expression was the same as one used to study the behavior of the differences between eigenvalues of certain random Hermitian matrices (Thomas) — a subject that Dyson was familiar with because such eigenvalues are used to represent the energy levels of heavy atomic nuclei!

Based on this revelation, Montgomery hypothesized that all the statistics of real zeros of $\xi(t)$ will match the corresponding statistics of eigenvalues of random Hermitian matrices. If this is true, then the xi function’s zeros—and by extension, the imaginary parts of some of the zeta function’s zeros—probably represent the energy levels of some physical object (Conrey, $349$).

The paper then goes on to discuss other aspects related to RH, such as a link to quantum chaos, that I suggest you read yourself.

There are quite a few other such references online, but I will just mention one more of Surprising connections between number theory and physics which contains some of the above noted details, as well as a few others related to RH.