The Riemann zeta function ζ(s) = Σ 1/n^s has an Euler product: ζ(s) = ∏ 1/(1 - p^(-s)) over primes p. Analytic continuation extends ζ to ℂ (except s=1), and the Riemann Hypothesis—all nontrivial zeros have Re(s) = 1/2—profoundly shapes prime distribution.
You already know that a series can converge or diverge, and that convergence depends on the rate at which terms shrink. The Riemann zeta function ζ(s) = 1/1ˢ + 1/2ˢ + 1/3ˢ + … begins as a series question: for which s does this sum converge? When s is a real number greater than 1, the terms decay fast enough and the series converges — this is the same p-series test from real analysis. For s = 1 you get the harmonic series, which diverges. So the half-plane Re(s) > 1 is where ζ(s) is initially defined.
The bridge to primes comes through the Euler product. Because every positive integer factors uniquely into primes (the fundamental theorem of arithmetic), the sum over all integers can be re-expressed as a product over all primes: ζ(s) = ∏ₚ 1/(1 − p⁻ˢ). This identity is not a coincidence — it is a direct analytic encoding of unique factorization. Each prime p contributes a geometric series 1 + p⁻ˢ + p⁻²ˢ + … to the product, and multiplying all these series together recovers the sum over all n⁻ˢ exactly once per integer. The Euler product is the first place where ζ(s) speaks directly about primes rather than integers.
The truly revolutionary step is analytic continuation. Riemann showed in 1859 that ζ(s), originally defined only for Re(s) > 1, can be extended to a meromorphic function on all of ℂ, with a single pole at s = 1. This is like discovering that a recipe valid for positive temperatures secretly works at negative temperatures too — with some interpretation. The extended function has obvious (trivial) zeros at s = −2, −4, −6, … and the remaining (nontrivial) zeros all lie in the critical strip 0 < Re(s) < 1. The functional equation ζ(s) = 2ˢπˢ⁻¹ sin(πs/2) Γ(1−s) ζ(1−s) shows the strip is symmetric around Re(s) = 1/2.
The Riemann Hypothesis conjectures that every nontrivial zero lies exactly on the line Re(s) = 1/2, called the critical line. Why does this matter for primes? The prime counting function π(x) (the number of primes up to x) can be written as an explicit formula involving sums over the nontrivial zeros of ζ. Each zero contributes an oscillatory term to this formula. If all zeros lie on Re(s) = 1/2, the oscillations are as small as possible, and the Prime Number Theorem's approximation π(x) ≈ x/ln(x) holds with the best possible error bound. Zeros off the critical line would create larger swings in prime distribution — in effect, "lumpiness" in the primes at large scales. The zeta function is thus the analytic lens through which the multiplicative structure of the integers becomes visible.