Rational or Not? This Basic Math Question Took Decades to Answer.

In June 1978, the organizers of a large mathematics conference in Marseille, France, announced a last-minute addition to the program. During the lunch hour, the mathematician Roger Apéry would present a proof that one of the most famous numbers in mathematics — “zeta of 3,” or ζ(3), as mathematicians write it — could not be expressed as a fraction of two whole numbers. It was what mathematicians call “irrational.”

Conference attendees were skeptical. The Riemann zeta function is one of the most central functions in number theory, and mathematicians had been trying for centuries to prove the irrationality of ζ(3) — the number that the zeta function outputs when its input is 3. Apéry, who was 61, was not widely viewed as a top mathematician. He had the French equivalent of a hillbilly accent and a reputation as a provocateur. Many attendees, assuming Apéry was pulling an elaborate hoax, arrived ready to pay the prankster back in his own coin. As one mathematician later recounted, they “came to cause a ruckus.”

The lecture quickly descended into pandemonium. With little explanation, Apéry presented equation after equation, some involving impossible operations like dividing by zero. When asked where his formulas came from, he claimed, “They grow in my garden.” Mathematicians greeted his assertions with hoots of laughter, called out to friends across the room, and threw paper airplanes.

But at least one person — Henri Cohen, now at the University of Bordeaux — emerged from the talk convinced that Apéry was correct. Cohen immediately began to flesh out the details of Apéry’s argument; within a couple of months, together with a handful of other mathematicians, he had completed the proof. When he presented their conclusions at a later conference, a listener grumbled, “A victory for the French peasant.”

Once mathematicians had, however reluctantly, accepted Apéry’s proof, many anticipated a flood of further irrationality results. Irrational numbers vastly outnumber rational ones: If you pick a point along the number line at random, it’s almost guaranteed to be irrational. Even though the numbers that feature in mathematics research are, by definition, not random, mathematicians believe most of them should be irrational too. But while mathematicians have succeeded in showing this basic fact for some numbers, such as π and e, for most other numbers it remains frustratingly hard to prove. Apéry’s technique, mathematicians hoped, might finally let them make headway, starting with values of the zeta function other than ζ(3).

“Everyone believed that it [was] just a question of one or two years to prove that every zeta value is irrational,” said Wadim Zudilin of Radboud University in the Netherlands.

But the predicted flood failed to materialize. No one really understood where Apéry’s formulas had come from, and when “you have a proof that’s so alien, it’s not always so easy to generalize, to repeat the magic,” said Frank Calegari of the University of Chicago. Mathematicians came to regard Apéry’s proof as an isolated miracle.

But now, Calegari and two other mathematicians — Vesselin Dimitrov of the California Institute of Technology and Yunqing Tang of the University of California, Berkeley — have shown how to broaden Apéry’s approach into a much more powerful method for proving that numbers are irrational. In doing so, they have established the irrationality of an infinite collection of zeta-like values.

Jean-Benoît Bost of Paris-Saclay University called their finding “a clear breakthrough in number theory.”

Mathematicians are enthused not just by the result but also by the researchers’ approach, which they used in 2021 to settle a 50-year-old conjecture about important equations in number theory called modular forms. “Maybe now we have enough tools to push this kind of subject way further than was thought possible,” said François Charles of the École Normale Supérieure in Paris. “It’s a very exciting time.”

Whereas Apéry’s proof seemed to come out of nowhere — one mathematician described it as “a mixture of miracles and mysteries” — the new paper fits his method into an expansive framework. This added clarity raises the hope that Calegari, Dimitrov and Tang’s advances will be easier to build on than Apéry’s were.

“Hopefully,” said Daniel Litt of the University of Toronto, “we’ll see a gold rush of related irrationality proofs soon.”

A Proof That Euler Missed

Since the earliest eras of mathematical discovery, people have been asking which numbers are rational. Two and a half millennia ago, the Pythagoreans held as a core belief that every number is the ratio of two whole numbers. They were shocked when a member of their school proved that the square root of 2 is not. Legend has it that as punishment, the offender was drowned.

The square root of 2 was just the start. Special numbers come pouring out of all areas of mathematical inquiry. Some, such as π, crop up when you calculate areas and volumes. Others are connected to particular functions — e, for instance, is the base of the natural logarithm. “It’s a challenge: You give yourself a number which occurs naturally in math, [and] you wonder whether it’s rational,” Cohen said. “If it’s rational, then it’s not a very interesting number.”

Many mathematicians take an Occam’s-razor point of view: Unless there’s a compelling reason why a number should be rational, it probably is not. After all, mathematicians have long known that most numbers are irrational.

Yet over the centuries, proofs of the irrationality of specific numbers have been rare. In the 1700s, the mathematical giant Leonhard Euler proved that e is irrational, and another mathematician, Johann Lambert, proved the same for π. Euler also showed that all even zeta values — the numbers ζ(2), ζ(4), ζ(6) and so on — equal some rational number times a power of π, the first step toward proving their irrationality. The proof was finally completed in the late 1800s.

But the status of many other simple numbers, such as π + e or ζ(5), remains a mystery, even now.

It might seem surprising that mathematicians are still grappling with such a basic question about numbers. But even though rationality is an elementary concept, researchers have few tools for proving that a given number is irrational. And frequently, those tools fail.

When mathematicians do succeed in proving a number’s irrationality, the core of their proof usually relies on one basic property of rational numbers: They don’t like to come near each other. For example, say you choose two fractions, one with a denominator of 7, the other with a denominator of 100. To measure the distance between them (by subtracting the smaller fraction from the larger one), you have to rewrite your fractions so that they have the same denominator. In this case, the common denominator is 700. So no matter which two fractions you start with, the distance between them is some whole number divided by 700 — meaning that at the very least, the fractions must be 1/700 apart. If you want fractions that are even closer together than 1/700, you’ll have to increase one of the two original denominators.

Flip this reasoning around, and it turns into a criterion for proving irrationality. Suppose you have a number k, and you want to figure out whether it’s rational. Maybe you notice that the distance between k and 4/7 is less than 1/700. That means k cannot have a denominator of 100 or less. Next, maybe you find a new fraction that allows you to rule out the possibility that k has a denominator of 1,000 or less — and then another fraction that rules out a denominator of 10,000 or less, and so on. If you can construct an infinite sequence of fractions that gradually rules out every possible denominator for k, then k cannot be rational.

Nearly every irrationality proof follows these lines. But you can’t just take any sequence of fractions that approaches k — you need fractions that approach k quickly compared to their denominators. This guarantees that the denominators they rule out keep growing larger. If your sequence doesn’t approach k quickly enough, you’ll only be able to rule out denominators up to a certain point, rather than all possible denominators.

There’s no general recipe for constructing a suitable sequence of fractions. Sometimes, a good sequence will fall into your lap. For example, the number e (approximately 2.71828) is equivalent to the following infinite sum:

$latex \frac{1}{1} + \frac{1}{1} + \frac{1}{2 \times 1} + \frac{1}{3 \times 2 \times 1} + \frac{1}{4 \times 3 \times 2 \times 1} + \cdots$.

If you halt this sum at any finite point and add up the terms, you get a fraction. And it takes little more than high school math to show that this sequence of fractions approaches e quickly enough to rule out all possible denominators.

But this trick doesn’t always work. For instance, Apéry’s irrational number, ζ(3), is defined as this infinite sum:

$latex \frac{1}{1^3} + \frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \cdots$.

If you halt this sum at each finite step and add the terms, the resulting fractions don’t approach ζ(3) quickly enough to rule out every possible denominator for ζ(3). There’s a chance that ζ(3) might be a rational number with a larger denominator than the ones you’ve ruled out.

Apéry’s stroke of genius was to construct a different sequence of fractions that do approach ζ(3) quickly enough to rule out every denominator. His construction used mathematics that dated back centuries — one article called it “a proof that Euler missed.” But even after mathematicians came to understand his method, they were unable to extend his success to other numbers of interest.

Like every irrationality proof, Apéry’s result instantly implied that a bunch of other numbers were also irrational — for example, ζ(3) + 3, or 4 × ζ(3). But mathematicians can’t get too excited about such freebies. What they really want is to prove that “important” numbers are irrational — numbers that “show up in one formula, [then] another one, also in different parts of mathematics,” Zudilin said.

Few numbers meet this standard more thoroughly than the values of the Riemann zeta function and the allied functions known as L-functions. The Riemann zeta function, ζ(x), transforms a number x into this infinite sum:

$latex \frac{1}{1^x} + \frac{1}{2^x} + \frac{1}{3^x} + \frac{1}{4^x} + \cdots$.

ζ(3), for instance, is the infinite sum you get when you plug in x = 3. The zeta function has long been known to govern the distribution of prime numbers. Meanwhile, L-functions — which are like the zeta function but have varying numerators — govern the distribution of primes in more complicated number systems. Over the past 50 years, L-functions have risen to special prominence in number theory because of their key role in the Langlands program, an ambitious effort to construct a “grand unified theory” of mathematics. But they also crop up in completely different areas of mathematics. For example, take the L-function whose numerators follow the pattern 1, −1, 0, 1, −1, 0, repeating. You get:

$latex \frac{1}{1^x} + \frac{-1}{2^x} + \frac{0}{3^x} + \frac{1}{4^x} + \frac{-1}{5^x} + \frac{0}{6^x} + \cdots$.

In addition to its role in number theory, this function, which we’ll call L(x), makes unexpected cameos in geometry. For example, if you multiply L(2) by a simple factor, you get the volume of the largest regular tetrahedron with “hyperbolic” geometry, the curved geometry of saddle shapes.

Mathematicians have been mulling over L(2) for at least two centuries. Over the years, they have come up with seven or eight different ways to approximate it with sequences of rational numbers. But none of these sequences approach it quickly enough to prove it irrational.

Researchers seemed to be at an impasse — until Calegari, Dimitrov and Tang decided to make it the centerpiece of their new approach to irrationality.

A Proof That Riemann Missed

In an irrationality proof, you want your sequence of fractions to rule out ever-larger denominators. Mathematicians have a well-loved strategy for understanding such a sequence: They’ll package it into a function. By studying the function, they gain access to an arsenal of tools, including all the techniques of calculus.

In this case, mathematicians construct a “power series” — a mathematical expression with infinitely many terms, such as 3 + 2x + 7x² + 4x³ + … — where you determine each coefficient by combining the number you’re studying with one fraction in the sequence, according to a particular formula. The first coefficient ends up capturing the size of the denominators ruled out by the first fraction; the second coefficient captures the size of the denominators ruled out by the second fraction; and so on.

Roughly speaking, the coefficients and the ruled-out denominators have an inverse relationship, meaning that your goal — proving that the ruled-out denominators approach infinity — is equivalent to showing that the coefficients approach zero.

The advantage of this repackaging is that you can then try to control the coefficients using properties of the power series as a whole. In this case, you want to study which x-values make the power series “blow up” to infinity. The terms in the power series involve increasingly high powers of x, so unless they are paired with extremely small coefficients, large x-values will make the power series blow up. As a result, if you can show that the power series does not blow up, even for large values of x, that tells you that the coefficients do indeed shrink to zero, just as you want.

To bring an especially rich set of tools to bear on this question, mathematicians consider “complex” values for x. Complex numbers combine a real part and an imaginary part, and can be represented as points in a two-dimensional plane.

Imagine starting at the number zero in the complex number plane and inflating a disk until you bump into the first complex number that makes your power series explode to infinity — what mathematicians call a singularity. If the radius of this disk is large enough, you can deduce that the coefficients of the power series shrink to zero fast enough to imply that your number is irrational.

Apéry’s proof and many other irrationality results can be rephrased in these terms, even though that’s not how they were originally written. But when it comes to L(2), the disk is too small. For this number, mathematicians viewed the power series approach as a dead end.

But Calegari, Dimitrov and Tang saw a potential way through. A singularity doesn’t always represent a final stopping point — that depends on what things look like when you hit the singularity. Sometimes the boundary of the disk hits a mass of singularities. If this happens, you’re out of luck. But other times, there might be just a few isolated singularities on the boundary. In those cases, you might be able to inflate your disk into a bigger region in the complex plane, steering clear of the singularities.

That’s what Calegari, Dimitrov and Tang hoped to do. Perhaps, they thought, the extra information contained in this larger region might enable them to get the control they needed over the power series’ coefficients. Some power series, Calegari said, can have a “wonderful life outside the disk.”

Over the course of four years, Calegari, Dimitrov and Tang figured out how to use this approach to prove that L(2) is irrational. “They developed a completely new criterion for deciding whether a given number is irrational,” Zudilin said. “It’s truly amazing.”

As with Apéry’s proof, the new method is a throwback to an earlier era, relying heavily on generalizations of calculus from the 1800s. Bost even called the new work “a proof that Riemann missed,” referring to Bernhard Riemann, one of the towering figures of 19th-century mathematics, after whom the Riemann zeta function is named.

The new proof doesn’t stop with L(2). We construct that number by replacing the 1s in the numerators of ζ(2) with a pattern of three repeating numbers: 1, −1, 0, 1, −1, 0 and so on. You can make an infinite collection of other ζ(2) variants with three repeating numerators — for instance, the repeating pattern 1, 4, 10, 1, 4, 10 …, which produces the infinite sum

$latex \frac{1}{1^2} + \frac{4}{2^2} + \frac{10}{3^2} + \frac{1}{4^2} + \frac{4}{5^2} + \frac{10}{6^2} + \cdots$.

Every such sum, the researchers proved, is also irrational (provided it doesn’t add up to zero). They also used their method to prove the irrationality of a completely different set of numbers made from products of logarithms. Such numbers were previously “completely out of reach,” Bost said.

Researchers anticipate that ζ(2) variants with repeating patterns of four numbers might be next. They’ve pinned their hopes, in particular, on proving the irrationality of “Catalan’s constant” — a variant with the repeating pattern 1, 0, −1, 0, 1, 0, −1, 0 …, which has been studied for more than 150 years.

“Catalan is so close,” Cohen said.

The results the team has achieved so far are “a proof of concept for the fact that their methods are able to go very, very far, much further than we expected a couple of years ago,” Charles said. “This is absolutely not the end of the story.”

After so many years spent peering through the fog, mathematicians are finally starting to clearly discern an array of landmarks in one of their most fundamental landscapes — the number line.