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.
Table of zeta function values at even integers
| s | ζ(s) | exact form |
|---|---|---|
| 2 | 1.64493… | π²/6 |
| 3 | 1.20206… | unknown (Apéry) |
| 4 | 1.08232… | π⁴/90 |
| 6 | 1.01734… | π⁶/945 |
| -2,-4,… | 0 | trivial zeros |
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 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.