The zeta function

Here’s a function that is super important in analytic number theory:

\zeta(s)=\displaystyle\sum_{n=1}^\infty n^{-s}=\frac1{1^s}+\frac1{2^s}+\frac1{3^s}+\cdots.

There is way more to be said about this function than I can possible fit in one, or even seventeen posts, so I won’t even attempt to talk about the Riemann hypothesis. Here is a fun fact though:

\zeta(s)=\displaystyle\sum_{n=1}^\infty n^{-s}=\prod_p\frac1{1-p^{-s}},

where the product is over all prime numbers p. How does this work? Well, each term \frac1{1-p^{-s}} is the sum of a geometric series. Expanding the sum, we get

\displaystyle\prod_p\frac1{1-p^{-s}}=(1+2^{-s}+2^{-2s}+\cdots)(1+3^{-s}+3^{-2s}+\cdots)\cdot\dots

If we were to expand the product of infinite sums, we need to take some entry from each sum in the product. Say we take the 3^{-s} term, the 5^{-3s} term, and the rest ones. When we multiply all of these together, we get 375^{-s}. We get each positive integer to show up exactly once (by unique factorization). Cool!

So what is \zeta(1)? According to the sum, it’s 1+\frac12+\frac13+\cdots. This sum does not converge. If you didn’t already know this, there are lots of proofs on the internet.

So if \zeta(1) is infinite, it better be infinite when we use the product definition. It’s obvious that a finite product of finite numbers is finite, so if \zeta(1) is going to be infinite, there had better be infinitely many primes.

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