The Riemann zeta-function is defined by \[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}, \quad \text{for $\mathrm{Re}(s) > 1$}, \] and the domain is extended to all of $\mathbb{C}$ by analytic continuation. The zeros of the Riemann zeta-function are the subject of the Riemann hypothesis.
Almost $50$ years ago, in April $1972$, Hugh Montgomery made an important conjecture about the pair correlation between these zeros. According to a famous anecdote, Freeman Dyson and Hugh Montgomery were having a casual exchange about their research, when they realised that they had obtained the same result for completely different problems. Montgomery had computed the pair correlation function of the zeros of $\zeta(s)$ and Dyson had earlier computed the pair correlation of the eigenvalues of random unitary matrices. Remarkably they found that the two were equal. Without elaborating this further, it essentially means that the zeros of the Riemann zeta function behave statistically like the eigenvalues of random unitary matrices.
This approach was taken a step further by Jon Keating and Nina Snaith when they modelled the zeta function using the characteristic polynomials of random unitary matrices [KS00b, KS00a]. Amongst other things they were able to make precise conjectures about the moments of the zeta function on the critical line. These ideas can be extended to more general $L$-functions.
Unfortunately these connections remain mostly conjectural. However, we can enter another world, the world of function fields, where they can be proved. In this world, instead of investigating the distribution of prime numbers, one is interested in understanding irreducible polynomials in $\mathbb{F}_q[t]$, where $\mathbb{F}_q$ denotes the field with $q$ elements. The Riemann hypothesis for the analogous zeta function in the context of $\mathbb{F}_q[t]$ is trivially true. More general $L$-functions can now not only be modelled by characteristic polynomials of unitary matrices but by virtue of Deligne's Theorem they actually are ratios of (scaled) characteristic polynomials of unitary matrices. As a result many conjectures in the world of the integers become facts in the world of function fields, as we will see below.
Denote by $\Lambda \colon \mathbb{Z} \rightarrow \mathbb{R}$ the von Mangoldt function \[ \Lambda(n) = \begin{cases} \log p, \quad &\text{if $n = p^k$ for $p$ a prime number} \\ 0, &\text{otherwise}. \end{cases} \] The Prime Number Theorem states that the number of prime numbers which are smaller than $x$, denoted by $\pi(x)$, satisfies $\pi(x) \sim \frac{x}{\log x}$ as $x \rightarrow \infty$. It is an elementary fact that this is equivalent to \[ \sum_{n \leq x} \Lambda(n) \sim x, \quad \text{as $x \rightarrow \infty$}. \] Similarly, understanding the count of primes in the interval $[x,x+H]$ is equivalent to understanding $\sum_{x \leq n \leq x+H} \Lambda(n)$. This can be shown to be asymptotically of size $H$ if $H>x^{1/2 + o(1)}$. In particular one might be interested to compute the variance of the number of primes in short intervals, that is \[ \mathrm{Var}(X,H) = \frac{1}{X}\int_2^X \left\lvert \sum_{x \leq n \leq x+H} \Lambda(n) - H \right\rvert^2 dx. \] Assuming the Riemann hypothesis and Montgomery's pair correlation conjecture, Goldston and Montgomery showed $\mathrm{Var}(X,h) \sim H(\log(X) - \log(h))$ if $H$ lies in the range $x^\delta < H < x^{1-\delta}$ for some $\delta >0$, see [GM87].
In the function field case we can analogously define a von Mangoldt function $\Lambda_{\mathbb{F}_q} \colon \mathbb{F}_q[t] \rightarrow \mathbb{R}$ via \[ \Lambda_{\mathbb{F}_q}(f) = \begin{cases} \deg P, \quad &\text{if $f = P^k$ for $P \in \mathbb{F}_q[t]$ irreducible} \\ 0, &\text{otherwise.} \end{cases} \] Similarly here, understanding the distribution of irreducible polynomials amounts to understanding sums of $\Lambda_{\mathbb{F}_q}$. The prime polynomial theorem is the equality \[ \sum_{\substack{\deg f = n \\ f \text{ monic}}} \Lambda_{\mathbb{F}_q}(f) = q^n, \] which is the exact analogue of the Prime Number Theorem. Jon Keating and Zeev Rudnick computed the variance of the number of prime polynomials in short intervals [KR14] using an equidistribution result due to Katz [Kat13] which guarantees that the unitary matrices appearing in certain $L$-functions on average behave like random unitary matrices. Having a result like this available turns complicated character sums into matrix integrals, which are usually straightforward to compute. Given $h < n-3$ they could show that the variance of $\Lambda_{\mathbb{F}_q}$ in intervals of length $q^{h+1}$ grows asymptotically like
\[
q^{h+1}(n-h-2), \quad \text{as $q \rightarrow \infty$}.
\]
Making the natural identifications
\[
q^{h+1} \leftrightarrow H, \quad h+1 \leftrightarrow \log H, \quad \text{and} \quad n+1 \leftrightarrow \log X,
\]
we can see that this corresponds to the asymptotic in the integer case.
During my MMath dissertation I looked at a more abstract version of this problem, and I think it is a nice example to demonstrate just how much we can show in the function field case as opposed to the number field case. In loose terms Galois representations $\rho$ of the function field $\mathbb{F}_q(t)$ encode information about the Galois extensions of $\mathbb{F}_q(t)$ and their arithmetic. In general, they are extremely rich and complicated to understand. Given such a $\rho$ one can define a so-called Artin $L$-function attached to $\rho$ and a generalised von Mangoldt function $\Lambda_\rho \colon \mathbb{F}_q[t] \rightarrow \mathbb{C}$. Using an equidistribution result by Will Sawin [Saw18)], I was able to compute the variance of $\Lambda_\rho$ in short intervals [Hoc21].
In the number field case there are corresponding results for Galois representations of $\mathbb{Q}$ [BKS16]. In fact, the results correspond qualitatively and quantitatively to the results in the function field setting that I obtained. However they are conditional on the generalised Riemann hypothesis as well as some other strong conjectures, which unfortunately seem far away from being proved any time soon.
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Having completed his MMath at Oxford, Leo is currently a PhD student at the University of Göttingen under the supervision of Damaris Schindler where he mainly concerns himself with problems related to counting rational points on varieties and Manin's conjecture. In addition he is working together on a project with Ezra Waxman where they try to use the results that Leo obtained in his MMath dissertation to answer questions related to Artin's primitve root conjecture over function fields.
References:
[BKS16] H. M. Bui, J. P. Keating, and D. J. Smith. On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for Lfunctions in the selberg class. Journal of the London Mathematical Society, 94(1):161–185, 2016.
[GM87] D. A. Goldston and H. L. Montgomery. Pair correlation of zeros and primes in short intervals. In Analytic number theory and Diophantine problems, pages 183–203. Springer, 1987.
[Hoc21] Leonhard Hochfilzer. Variance of sums in short intervals and L-functions in Fq[t]. Journal of Number Theory, 2021.
[Kat13] N. M. Katz. Witt Vectors and a Question of Keating and Rudnick. International Mathematics Research Notices, 2013(16):3613–3638, 06 2013.
[KR14] J. P. Keating and Z. Rudnick. The variance of the number of prime polynomials in short intervals and in residue classes. International Mathematics Research Notices, 2014(1):259–288, 2014.
[KS00a] J. P. Keating and N. C. Snaith. Random matrix theory and L-functions at s = 1/2. Communications in Mathematical Physics, 214(1):91–100, 2000.
[KS00b] J. P. Keating and N. C. Snaith. Random matrix theory and ζ(1/2 + it). Communications in Mathematical Physics, 214(1):57–89, 2000.
[Saw18] W. Sawin. The equidistribution of L-functions of twists by Witt vector Dirichlet characters over function fields. arXiv preprint arXiv:1805.04330, 2018.