I recently wrote a semester paper on probabilistic number theory. I’m very grateful to both of my supervisors, Dr. Vivian Kuperberg and Prof. Dr. Emmanuel Kowalski, for their insights and suggestions. Here’s the abstract:

This report is an exposition of two central limit theorems in probabilistic number theory. We begin by introducing preliminary results from number theory and probability theory. Then we prove the Erdős-Kac theorem for the asymptotic behaviour of the prime divisor counting function. The majority of the report is devoted to Radziwiłł and Soundarajan’s recent proof of the Selberg central limit theorem for $\log |\zeta(\frac{1}{2} + it)|$.

You’re most welcome to have a look at the actual semester paper. Here I’d like to give a more informal discussion of its contents.

“Deterministic” number theory #

A central theme in analytic number theory is deriving estimates for averages over arithmetic functions, e.g. something like

where $f: \mathbb{N} \to \mathbb{C}$ is some arithmetic function. Many “named theorems” are estimates of the above form. For example:

• If we let $f(n) = x^{-1} 1_P(n)$, where $1_P$ is the indicator of the set of prime numbers, then we get Mertens’ second theorem.
• If instead $f(n) = d(n)$, the function counting the number of divisors of a positive integer $n$, then a theorem of Dirichlet asserts that

• Actually, the Prime number theorem is equivalent to an estimate of the above form. Just take $f$ to be the von Mangoldt function $\Lambda$, defined by

Probabilistic number theory #

In my semester paper, I focused on two central limit theorems from probabilistic number theory: the Erdős–Kac and Selberg central limit theorems. Here, the approach is slightly different.

Sums and integrals as expectations #

First, we let $f: \Omega \to \mathbb{C}$ be an arithmetically defined quantity more generally; typically $\Omega = \mathbb{N}$ or $\mathbb{R}$. So we could take $f$ to be an arithmetic function, but now we also allow for functions of arithmetic functions. For example, if we take $g: \mathbb{N} \to \mathbb{C}$ to be an arithmetic function and post-compose with $x \mapsto x^2 / \sqrt{\log \log x}$, then we could have $f: \mathbb{N} \to \mathbb{C}$ be a weird-looking expression defined by

To highlight the connection with probability theory, fix a positive integer $x$ and let $U_x$ be a random variable uniformly distributed on $\\{1, ..., x\\}$. Then we obtain

So, estimating $x^{-1} \sum_{n \le x} f(n)$ is the same as estimating an expectation.

Similarly, the problem of estimating the integral

also comes down to estimating an expectation. In the continuous case, let $x$ be an arbitrary positive real number and take $U_x$ to be a random variable uniformly distributed over $[0, x]$. Then the above integral is precisely $\mathbb{E}(f(U_x))$.

Natural questions #

This naturally prompts the following questions:

• Could we also estimate the variance of $f(U_x)$?
• Could we even say something about the asymptotic distribution of $f(U_x)$ as $x \to \infty$?

Ideally, we can prove something like this:

“Let $U_x: \Omega \to \Omega_x$ denote a random variable uniformly distributed on $\Omega_x := \\{1, ..., x\\}$ (or $[0, x]$). Let $Y$ be a random variable with some simple distribution. Then as $x \to \infty$, we have

where $\xrightharpoonup{}$ denotes convergence in distribution.”

A remarkable fact is that some rather convoluted expressions involving arithmetic functions can converge in distribution to standard normal random variables as $x \to \infty$.

General proof strategy #

Proving results of this form involves machinery from both number theory and probability. We typically use number theory to massage the expression $f(U_x)$, so it can be approximated by some simpler non-arithmetic random variable. This done, we can use limit theorems from probability to finish off the proof.

The Erdős-Kac theorem #

This serves as a good “toy example” for the proof strategy outlined above. Also, it’s not too difficult deriving a heuristic proof; for details, see page 14.

First, some notation. Let $\omega$ denote the function indicating the number of distinct prime divisors of a positive integer. For example, $\omega(6) = 2$ since $6 = 2 \cdot 3$, while $\omega(9) = 1$, since $9 = 3^2$. Then we have:

Theorem. (Erdős-Kac, 1940). Let $U_n$ be a random variable uniformly distributed on $\{1, ..., n\}$ and define

Then the sequence $(W_n)_{n \ge 1}$ of random variables converges in distribution to a standard normal random variable as $n \to \infty$.

Selberg’s central limit theorem for $\log |\zeta(1/2 + it)|$ #

Let $\zeta$ denote the Riemann zeta function, defined by

The Riemann hypothesis, perhaps the most famous open problem in mathematics, says that the only zeros of the zeta function are of the form $s = -2k$ for positive integers $k$ or lie on the critical strip $\\{1/2 + it : t \in \mathbb{R}\\}$.

Now fix a positive number $T > 0$, and consider the random variable

where $U_t$ is a random variable uniformly distributed on the interval $[-T, T]$. Considering just how hard the Riemann hypothesis is, it’s pretty surprising that the behaviour of $L_T$ is relatively well-understood. In fact, we have:

Theorem. (Selberg, 1946). Let $L_T$ be the random variable defined as above, and set

Then the sequence $(M_T)_{T \ge 0}$ of random variables converges in distribution to a standard normal random variable as $T \to \infty$.

In my semester paper, I cover the 2016 proof due to Radziwiłł and Soundarajan. Just as in the Erdős-Kac theorem, one can get an intuitive sense of why this result is plausible.

Final words #

For details, I encourage you to have a look at the actual paper. I hope you enjoy reading it as much as I enjoyed writing it.