# Riemann Hypothesis

**Type:** Conjecture  
**Domain:** Named Results · Number Theory  
**Proposed:** 1859, Bernhard Riemann  
**Status:** Unsolved  
**Codex URL:** /mathematics/named-results/riemann-hypothesis/  
**Entry status:** Live — v1.0 (2026-05-27)

## Statement
Every non-trivial zero of the Riemann zeta function ζ(s) has real part exactly 1/2.

## Summary
Millennium Prize Problem ($1M, Clay Mathematics Institute). Controls the precise distribution of primes via Riemann's explicit formula. Verified numerically for the first 10¹³ zeros; not proven for all.

## The zeta function
ζ(s) = Σ 1/nˢ (Re(s)>1), extended by analytic continuation. Trivial zeros at s=−2,−4,−6,.... Non-trivial zeros lie in critical strip 0<Re(s)<1.

## Partial results
- Hardy (1914): infinitely many zeros on critical line
- Levinson (1974): ≥1/3 of zeros on critical line
- Conrey (1989): >40% of zeros on critical line

## Sources
### Tier 1
- Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse.
- Titchmarsh, E.C. (1986). *The Theory of the Riemann Zeta-Function*. 2nd ed.
### Tier 2
- Bombieri, E. (2000). Clay Mathematics Institute Millennium Problem description.
### Tier 3
- Derbyshire, J. (2003). *Prime Obsession*. Joseph Henry Press.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*