chapter 04

The zeta function

One input, one infinite sum, and an unexpected amount of information.

Turn the exponent into an input

For real numbers $s>1$, define the Riemann zeta function by

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

A function is simply a rule: supply $s$, receive a value. For example, $\zeta(2)=\pi^2/6\approx1.645$. Increasing $s$ crushes every term after the initial 1, so $\zeta(s)$ approaches 1.

Some values on the safe side of $s=1$.

$s$$\zeta(s)$ 21.644934… 31.202056… 41.082323… 101.000994…

A warning at $s=1$

At $s=1$, zeta becomes the divergent harmonic series. As $s$ approaches 1 from the right, $\zeta(s)$ shoots upward. We say zeta has a pole at 1.

The series definition also stops converging when the real part of $s$ is at most 1. But this does not mean the function itself must stop there—just as the expression $(x^2-1)/(x-1)$ can simplify to $x+1$ away from its apparent problem at $x=1$.

A formula can outgrow its first definition

Through analytic continuation, mathematicians extend zeta to nearly the entire complex plane while preserving the values already defined for $s>1$. The only unavoidable exception is the pole at $s=1$.

This extension is not arbitrary. Complex differentiability is so rigid that, once an extension exists, it is unique.