ᴛʜᴇ ɢᴏꜱʜ ᴘᴀʀᴛɪᴄʟᴇ

Most famous constants announce themselves immediately. For example, π appears wherever circles show up, e emerges from growth and calculus, and so on. But there's a stranger hiding deep within this infinite series:

ζ(3) = 1 + 1 / 2^3 + 1 / 3^3 + 1 / 4^3 + . . .

The number this sum converges to is called Apéry's constant, and despite looking innocent, it resisted proof for centuries. Mathematicians strongly suspected that it was irrational but nobody could prove it until 1978.

The Zeta Function Apéry's constant comes from one of the most important objects in mathematics: the Riemann zeta function.

For real numbers s > 1,

ζ(s) = ∑_{n=1}^∞ 1 / n^s

(the infinite series of 1 / 1 ^ s + 1 / 2 ^ s + 1 / 3 ^ s + . . . )

At first glance, this is just an infinite sum. But the zeta function secretly connects prime numbers, complex analysis, quantum physics, probability, cryptography, and the distribution of the primes themselves.

Some values are beautifully understood. For example,

ζ(2) = π^2 / 6

ζ(4) = π^4 / 90

In fact, every even positive integer produces a formula involving powers of π. But the odd inputs are another story.

Nobody knows a comparably elegant formula for

ζ(3), ζ(5), ζ(7), . . .

These numbers are mysterious, and ζ(3) became the first "battleground".

Roger Apéry's Bombshell

Numerically, Apéry's constant equals approximately

1.202056903159694...

The question sounds deceptively simple: Is this number rational? For over 200 years, nobody knew. That's remarkable because Euler had solved the analogous problem for ζ(2) in the 1700s.

In 1978, French Mathematician Roger Apéry announced that ζ(3) was irrational in a lecture. The announcement was met with criticism in part because Apéry was relatively unknown at the time. He also gave the lecture in French, made jokes throughout, and omitted several key explanations needed to follow the proof..

For example, there was an equation at the beginning of his lecture that no one knew but formed the core of his proof. When asked where this equation came from, Apéry is said to have answered "They grow in my garden," which was said to have caused many in the audience to stand up and leave the room.

However, someone in attendance had an electronic calculator (uncommon at the time) and with a short program, checked Apéry's equation and found it correct.

The equation in question is below, which was an unknown series representation of ζ(3) at the time:

With this expression, he was able to use a condition for irrationality that German mathematician Gustav Lejeune Dirichlet had derived in the 19th century. It states that a number χ is irrational if there are an infinite number of integers p and q, so that the following inequality is satisfied:

Here, c and δ denote constant values. Although the formula looks complicated, it basically means that χ can be approximated by fractions, but there is no fractional number that corresponds to χ exactly. Apéry succeeded in deriving this inequality for ζ(3), and thus the number is irrational.

In simpler terms, Apéry's proof constructed two sequences of integers:

a_n and b_n

such that

a_n / b_n

approximate ζ(3) far too well for a rational number.

This is the key philosophical idea. If a number is rational, there are limits to how accurately fractions can approximate it without eventually becoming exact. Apéry built approximations that violated those limits.

The machinery involved strange recursive sequences and combinatorial identities that seemed to come out of nowhere.

Even now, many mathematicians describe the proof as "magical".

To honor his work, the value of ζ(3) now bears his name and is known as Apéry's constant. This doesn't answer all of the questions associated with the number, however. We are still looking for a clear numerical value for ζ(3) that can be expressed with known constants, much in the same way ζ(2) is.

But regardless, Apéry's constant feels like an accident. It emerges from a simple series, has no known simple closed form, and required centuries to understand even partially. And yet it keeps appearing across math and physics like a recurring character in a story nobody fully understands.