the Euler product
/ OY-ler /
The Euler product is the single identity that turns the zeta function from a curiosity about sums into the master key of number theory: for Re s > 1, zeta(s) = sum over n of 1/n^s equals a product over the primes, product over primes p of 1/(1 - p^(-s)). The left side runs over all positive integers; the right side runs only over the primes 2, 3, 5, 7, 11, ... . Euler's astonishing discovery is that these two completely different-looking expressions are equal.
Why is it true? Each factor is a geometric series in disguise: 1/(1 - p^(-s)) = 1 + p^(-s) + p^(-2s) + p^(-3s) + ... . When you multiply these geometric series over ALL primes and expand, a typical term in the product picks one power p^(-k s) from each prime's factor and multiplies them, giving (p_1^(a_1) p_2^(a_2) ...)^(-s) = 1/n^s for some integer n. The fundamental theorem of arithmetic — every integer factors into primes in exactly one way — guarantees each 1/n^s appears exactly once. So the product, expanded, is precisely the sum. The Euler product IS the unique-factorisation theorem, written analytically.
The consequences are enormous. Because none of the factors 1/(1 - p^(-s)) is zero and the product converges for Re s > 1, zeta(s) has no zeros in that half-plane — a fact that immediately gives Euler's analytic proof that there are infinitely many primes (if there were finitely many, the product would be a finite number even at s = 1, contradicting the pole). More deeply, taking the logarithm of the Euler product links the zeros of zeta to the distribution of primes, the engine behind the prime number theorem. A caution: the product equals zeta only where it converges, Re s > 1; the prime structure must be read off through analytic continuation to reach the critical strip.
Check the first factors at s = 2, where zeta(2) = pi^2/6 = 1.6449... . The product 1/(1 - 1/4) * 1/(1 - 1/9) * 1/(1 - 1/25) * 1/(1 - 1/49) * ... = (4/3)(9/8)(25/24)(49/48)... climbs toward 1.6449 as more primes are included. Just the primes 2, 3, 5, 7 already give 1.5 * 1.125 * 1.0417 * 1.0208 = 1.832 * ... converging steadily to pi^2/6.
Each prime contributes one geometric-series factor; unique factorisation is exactly why the product equals the sum.
The Euler product converges only in the half-plane Re s > 1. It does NOT extend to the critical strip, so you cannot read zeta's interesting zeros directly from it — those live where the product has broken down and only the continued function survives.