chapter 05

Euler’s product

The bridge from an infinite sum over integers to an infinite product over primes.

The primes were inside all along

Euler discovered that for $s>1$, the zeta sum can also be written as a product with one factor for every prime:

$$\zeta(s)=\prod_{p\ \mathrm{prime}}\frac1{1-p^{-s}}=\frac1{1-2^{-s}}\frac1{1-3^{-s}}\frac1{1-5^{-s}}\cdots.$$

This is the hinge on which the entire story turns. The left side mentions every positive integer. The right side mentions only primes.

Why the product works

Each prime factor expands as a geometric series:

$$\frac1{1-p^{-s}}=1+\frac1{p^s}+\frac1{p^{2s}}+\frac1{p^{3s}}+\cdots.$$

Multiply one such series for every prime. Choosing $1/2^{as}$ from the 2-series, $1/3^{bs}$ from the 3-series, and so on produces

$$\frac1{(2^a3^b5^c\cdots)^s}.$$

Unique prime factorization guarantees that every positive integer appears exactly once. The multiplication therefore reconstructs $1+2^{-s}+3^{-s}+\cdots$.

$2^2\cdot3$$12$ $2\cdot5^2$$50$ $3\cdot7$$21$

A first fact about zeros

When the real part of $s$ is greater than 1, every Euler factor is finite and nonzero, and the product converges. Consequently, $\zeta(s)$ cannot be zero there.

To find zeros—the inputs where zeta vanishes—we must leave this safe half-plane and use the extended function.