The Riemann zeta function is ζ(s) = 1 + 1/2ˢ + 1/3ˢ + 1/4ˢ + ⋯ Euler studied the real version and found ζ(2) = π²/6 (the Basel problem) and the product formula ζ(s) = ∏ 1/(1-p⁻ˢ) over all primes. Riemann extended the function to complex numbers in his landmark 1859 paper.
Riemann's key insight: extending ζ(s) to complex s, the non-trivial zeros (where ζ(s) = 0 with 0 < Re(s) < 1) control the distribution of prime numbers. Each zero contributes an oscillation to the prime-counting function. Riemann conjectured in 1859 that all non-trivial zeros lie on the line Re(s) = 1/2. This is the Riemann Hypothesis.
Over 10 trillion non-trivial zeros have been verified to lie on Re(s) = 1/2. No counterexample has ever been found. The Clay Mathematics Institute offers $1 million for a proof (or disproof). A proof would give the sharpest possible bound on prime distribution errors. The Riemann Hypothesis has been unproved for 165 years.
The sum on the left treats every integer equally. The product on the right runs over primes only. They are equal: a consequence of unique prime factorisation. Every integer n = p₁^a₁ × p₂^a₂ × ⋯ contributes exactly once to both expressions.
The Riemann zeta function satisfies a symmetry: zeta(s) = 2^s * pi^(s-1) * sin(pi*s/2) * Gamma(1-s) * zeta(1-s). This extends zeta to all complex numbers s (except s = 1) and relates the value at s to the value at 1-s. It shows the non-trivial zeros come in pairs: if s is a zero, so is 1-s. The trivial zeros at s = -2, -4, -6, ... arise from the sin(pi*s/2) factor.
The Riemann zeta function is zeta(s) = 1 + 1/2^s + 1/3^s + ... Euler evaluated it at even integers: zeta(2) = pi^2/6, zeta(4) = pi^4/90. Riemann extended it to complex s in 1859 and conjectured all non-trivial zeros lie on Re(s) = 1/2. This Riemann Hypothesis is unproved after 165 years and is a Clay Millennium Prize problem worth $1 million. Over 10 trillion zeros have been verified on the critical line. The zeros control prime distribution: each zero contributes an oscillation to the prime-counting function.