It’s been 162 years since Bernhard Riemann posed a seminal question about the distribution of prime numbers. Despite their best efforts, mathematicians have made very little progress on the Riemann hypothesis. But they have managed to make headway on simpler related problems.

In a paper posted in September, Paul Nelson of the Institute for Advanced Study has solved the subconvexity problem, a kind of lighter-weight version of Riemann’s question. The proof is a significant achievement on its own and teases the possibility that even greater discoveries related to prime numbers may be in store.

“It’s a bit of a far-fetched dream, but you could hyper-optimistically hope that maybe we get some insight in how the [Riemann hypothesis] works by working on problems like this,” Nelson said.

The Riemann hypothesis and the subconvexity problem are important because prime numbers are the most fundamental — and most fundamentally mysterious — objects in mathematics. When you plot them on the number line, there appears to be no pattern to how they’re distributed. But in 1859 Riemann devised an object called the Riemann zeta function — a kind of infinite sum — which fueled a revolutionary approach that, if proved to work, would unlock the primes’ hidden structure.

“It proves a result that a few years ago would have been regarded as science fiction,” said Valentin Blomer of the University of Bonn.

Getting Complex

Riemann’s question hinges on the Riemann zeta function. The terms it adds together are the reciprocals of the whole numbers, in which the denominators are raised to a power defined by a variable, s (so $latex frac{1}{1^{s}}$, $latex frac{1}{2^{s}}$, $latex frac{1}{3^{s}}$ and so on).

Riemann proposed that if mathematicians could prove a basic property of this function — what it takes for it to equal zero — they’d be able to estimate with great accuracy how many prime numbers there are along any given interval on the number line.

Prior to Riemann, Leonhard Euler constructed a similar function and used it to create a new proof that there are infinitely many primes. In Euler’s function, the denominators are raised to powers that are real numbers. The Riemann zeta function, by contrast, assigns complex numbers to the variable s, an innovation that brings the whole vast store of techniques from complex analysis to bear on questions in number theory.

Complex numbers have two parts, one real and one imaginary, the latter of which relates to the imaginary number i, defined as the square root of −1. Examples include 3 + 4i and 2 − 6i. In these cases, the 3 and the 2 are the real parts, while the 4i and −6i are the imaginary parts.

The Riemann hypothesis is about which values of s make the Riemann zeta function equal zero. It predicts that the only important, or nontrivial, values of s that do this are complex numbers whose real part equals $latex frac{1}{2}$. (The function also equals zero whenever s is a negative even integer with an imaginary part that equals zero, but those zeros are easy to see and are considered trivial.) If the Riemann hypothesis is true, the Riemann zeta function explains how primes are distributed on the number line. (Exactly how it explains that is c