Transcription
Take any positive whole number, cube it, then take one divided by that cube. So, for one, you get one over one, which is just one. For two, you get one over eight. For three, one over 27. For four, one over 64.
Now, add all of them up, every single one, for every positive integer, stretching out forever. You might expect this to blow up to infinity, and that's a reasonable instinct, because after all, you're adding infinitely many positive things together. But, watch what happens if you actually start tallying. After 10 terms, you're at about 1.20. After 100 terms, about 1.202. After 1,000, you're at 1.20205, and the digits are barely budging.
This sum converges, >> [music] >> and it converges to something very specific, a number that begins 1.2020569, and just keeps going. Mathematicians call this number zeta of three, or Apéry's constant, and it is, in a precise sense, one of the strangest constants [music] in all of mathematics.
Now, you might wonder why this particular number deserves any special attention. We have pi, we have e, we have the golden ratio, constants that show up everywhere, and have clean, elegant identities attached to them. And the truth is, zeta of three does show up in real physics in ways that matter for actual measurements of the real world. But, what makes it strange isn't where it appears, it's what we don't know about it. To appreciate that, though, we need to take a short detour through the problem's more famous sibling. [music]
Instead of summing the reciprocals of cubes, sum the reciprocals of squares. 1 + 1/4 + 1/9 + 1/16, and so on. This was the famous Basel problem, posed in the 1600s, [music] and it tormented mathematicians for nearly a century, until Euler, in 1735, showed that the answer is pi squared over six. That result is shocking. You take these reciprocal squares, [music] objects that feel like they belong purely to arithmetic, and you get pi, a number that lives in the world of circles and geometry. >> [music] >> Where did the circle come from?
There's a particularly beautiful way to see this connection, due to a mathematician named Beukers, and it involves a double integral. Consider the unit square. X goes from zero to one, Y goes from zero to one. Now, you might be tempted to just integrate 1 over 1 minus XY over that square. >> [music] >> And formally, if you expand 1 over 1 minus XY as a geometric series, 1 + XY + X squared Y squared, and so on, and integrate each term, you get exactly 1 1/4 1/9 1/16. Each term peels off one reciprocal square, so the series adds up to zeta of two. But, there's a catch. The function 1 over 1 minus XY blows up at the corner where X and Y are both one, and it blows up badly enough that the integral itself actually diverges. The term-by-term calculation is seductive, but you can't just integrate the function as is over the whole square.
What Beukers actually used is a more subtle integral. You integrate negative log of XY divided by 1 minus XY over the unit square. That extra factor of negative log XY tames the singularity at the corner. It vanishes fast enough there to make the whole integral converge. And when you work it out, this integral equals not zeta of two, but twice zeta of two, pi squared over three. >> [music] >> The logarithm is doing real work here, not just smoothing things out, but doubling the arithmetic content. And from there, with the right change of variables, rotating coordinates, exploiting symmetry, you can massage that integral into something where pi reveals itself. The circle sneaks in through the evaluation of trigonometric integrals [music] that arise after a rotation-like change of variables. It's gorgeous, and it gives you this feeling that the connection between the sum and pi is not an accident, but something deeply geometric. >> [music] >>
So, here's the natural question. Can you do the same thing for cubes? You might try a triple integral. Integrate 1 over 1 minus XYZ over the unit cube, X, Y, Z, all from zero to one. There's a singularity at the corner where all three variables equal one, but it turns out the singularity in three dimensions is mild enough that the integral actually converges. And the geometric series trick still works. Expand, integrate term by term, and you recover zeta of three. So far, so good. But, now try to evaluate that integral the way Beukers did for the square. Try rotating, try exploiting symmetry, try pulling out a pi, and you hit a wall. >> [music] >> The three-dimensional geometry of the unit cube simply does not cooperate the same way. The symmetries that let you crack the two-dimensional case, where you could cleverly decompose the domain into pieces that each evaluate to something involving pi, those symmetries don't lift cleanly into three dimensions. The obstruction isn't a matter of not being clever enough. It's that the integral genuinely resists closed-form evaluation in terms of pi, or any other known constant. >> [music] >>
And this is what makes zeta of three so unusual. For even powers, zeta of two, zeta of four, zeta of six, all the way up, Euler found beautiful closed forms, all rational multiples of powers of pi. Zeta of four is pi to the fourth over 90. Zeta of six is pi to the sixth over 945. There's a uniform pattern, and it's been proven to hold for every even positive integer. But, for odd values, zeta of three, zeta of five, zeta of seven, nobody has ever found a closed form, not [music] for a single one of them.
For a long time, mathematicians didn't even know whether zeta of three was irrational. It certainly looks irrational if you stare at its decimal expansion, but looking irrational and proving irrationality are two very different things. That's where Roger Apéry enters the story. In 1978, Apéry was 61 years old, not particularly famous, and he stood up at the Journées Arithmétiques in Luminy, near Marseille, to announce that he had proven zeta of three is irrational. The reception was, to put it mildly, skeptical. His proof involved a pair of rapidly converging sequences of rational numbers that approached zeta of three, and the key was showing they approached it too quickly [music] for zeta of three to be rational. The approximations were too good, in a sense made precise by a classical irrationality criterion. [music] But, the sequences themselves looked like they had been pulled from thin air. The recurrence relations defining them were complicated, and Apéry offered very little motivation for why anyone should have guessed them. Several mathematicians in the audience openly doubted the proof. Henri Cohen, along with colleagues, including Alfred van der Poorten, went and checked it line by line, and to everyone's astonishment, it worked. [music] Every step held.
What's remarkable, and what still feels a bit like magic nearly 50 years later, is that nobody has found a truly natural explanation for why Apéry's sequences work. People have reinterpreted the proof in terms of modular forms, in terms of differential equations, in terms of periods of algebraic varieties, but none of these perspectives make you feel like you could have discovered those sequences yourself. There's something hiding inside zeta of three that we haven't fully uncovered.
And yet, this seemingly inscrutable constant shows up in the real, physical, measurable world. When physicists compute the magnetic moment of the electron, how strongly it behaves as a tiny magnet, they use quantum electrodynamics, QED, which builds the answer as a series of corrections. The first correction, [music] the famous Schwinger term, is alpha over 2 pi, where alpha is the fine structure constant. It [music] comes from a single Feynman diagram with one loop. The second correction involves two loops, and is already much more complicated. But, it's the third-order correction, involving three-loop diagrams, 72 of them, where zeta of three appears explicitly in the coefficient. It's not an approximation. It's not a numerical coincidence. The analytical evaluation of those diagrams produces Apéry's constant as an exact factor. And this matters because the electron's anomalous magnetic moment is one of the most precisely measured quantities in all of science. Theory [music] and experiment agree to something like 10 to 12 significant digits. Zeta of three is baked into that agreement. If you removed it, if you replaced it with some nearby rational number, the prediction would be wrong. A number that comes from the purely arithmetic question of summing reciprocal cubes turns out to be a structural constant of quantum field theory.
So, where does this leave us? >> [music] >> We know zeta of three is irrational. We know it appears in physics, but we don't even know if it's transcendental. We don't have a closed form. Don Zagier and others have conjectured deep connections between values of zeta functions at odd integers and something called motivic cohomology, a framework where these constants should be, in some precise algebraic geometric sense, fundamental building blocks, periods of mixed Tate motives. If those conjectures are right, the reason we can't write zeta of three in terms of pi isn't that we haven't been clever enough. It's that zeta of three is a genuinely independent constant, as fundamental as pi, but linearly independent from any power of pi over the rationals. It's not a complicated expression in known constants. [music] It's a new constant. We just don't have the language for it yet.
Think back to that sum we started with. 1 + 1/8 + 1/27 + 1/64. >> [music] >> It looked so simple, almost boring, just shrinking fractions added up. But, the number they converge to sits at the intersection of number theory, geometry, and quantum physics. And after centuries of work, it still refuses to tell us exactly what it is. Sometimes the simplest questions are the ones that take the longest to answer. And sometimes they haven't been answered yet. >> [music]