John Carlos Baez on Nostr: I love fun little facts like this. It may seem mysterious at first, but it's really ...

I love fun little facts like this. It may seem mysterious at first, but it's really just basic algebra!

To see this fact, take the Riemann zeta function

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

and square it out:

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

= 1/1ˢ + 1/2ˢ + 1/3ˢ + 1/4ˢ+ ...
+1/2ˢ + 1/4ˢ + 1/6ˢ + 1/8ˢ+ ...
+1/3ˢ + 1/6ˢ + 1/9ˢ + 1/12ˢ+ ...
+1/4ˢ + 1/8ˢ + 1/12ˢ + 1/16ˢ+ ...

Notice you get 1/1ˢ just once here, but 1/2ˢ twice, and 1/3ˢ twice, and 1/4ˢ three times, and... see the pattern?

You get a term equal to 1/nˢ for each way you can write n as a product of two natural numbers! So you get the number of divisors of n times 1/nˢ.