The Euler Product
Euler made a discovery that still feels like magic: an infinite sum over all whole numbers
can be rewritten as an infinite product over just the primes. This Euler
product is the doorway from the discrete world of
multiplicative functions
into analytic number theory — it turns
unique factorisation
into an equation about infinite series.
The prototype
Consider the sum of 1/n^{s} over all positive integers (for
s > 1 so it converges). Euler's identity is:
\sum_{n=1}^{\infty} \frac{1}{n^{s}} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}.
The left side is a sum over every number; the right is a product over primes only.
That they are equal is one of the most consequential equations in mathematics — the function on
either side is the
Riemann zeta function.
Why it holds: factorisation in disguise
Expand each factor as a geometric series,
\frac{1}{1 - p^{-s}} = 1 + p^{-s} + p^{-2s} + \cdots, and multiply them
all out. Each term of the product picks one power of each prime — and multiplies to
1/n^{s} for a number n with that exact prime
factorisation. Because every n has one factorisation,
each 1/n^{s} appears exactly once:
\prod_p \big(1 + p^{-s} + p^{-2s} + \cdots\big) = \sum_n \frac{1}{n^{s}}.
The Euler product is unique factorisation, rephrased analytically.
A first stunning consequence
Set s = 1. The sum \sum 1/n (the harmonic
series) diverges to infinity — so the product over primes must diverge too. But that can
only happen if there are infinitely many primes — a brand-new, analytic proof of
Euclid's theorem. Pushed further, the same idea shows
\sum_p 1/p diverges, quantifying just how "dense" the primes are and
leading straight to the
Prime Number Theorem.