The Bernoulli numbers are defined like this:

x/(eˣ - 1) = B₀ + B₁x + B₂x²/2! + B₃x³/3! + ....

and if you grind them out, you get

B₀ = 1
B₁ = -1/2
B₂ = 1/6
B₃ = 0
B₄ = -1/30

and so on. The pattern is sort of strange.

They're connected to hundreds of interesting things. For example if you want to figure out a sum like

1³ + 2³ + 3³ + ... + 10³

or

1⁸ + 2⁸ + 3⁸ + ... + 1,000,000⁸

you can use a formula that involves Bernoulli numbers. The video here explains it.

But where the hell did this function x/(eˣ - 1) come from?

If D means derivative:

(Df)(x) = f'(x)

then eᴰ - 1 is a so-called 'difference operator':

((eᴰ - 1)f)(x) = f(x+1) - f(x)

which you can show using the Taylor series for f. So D/(eᴰ - 1) is about derivatives versus differences, and its inverse is about integrals versus sums. This lets you reduce sums like those above to integrals... 𝑖𝑓 you know your Bernoulli numbers.

But x/(eˣ - 1) also shows up when you compute the expected energy of a quantum harmonic oscillator in thermal equilibrium!

Let's work in units where ℏ = 1. Say we have a quantum harmonic oscillator whose allowed energies are 0, 1, 2, 3, ... etcetera. If we take its average or 'expected' energy at temperature T, and divide it by T, we get

x/(eˣ - 1)

where x = 1/T.

So the quantum harmonic oscillator secretly knows about Bernoulli numbers.

What does this fact really mean??? I don't know. I once read a book called Triangle of Thought about a conversation between Alain Connes and two other mathematicians, and he said this fact explained a lot of stuff. But he didn't go into any detail, so I'm left looking for clues.

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https://www.youtube.com/watch?v=fw1kRz83Fj0