Number TheoryUnsolved
Riemann Hypothesis
Do all non-trivial zeros of the Riemann zeta function have real part equal to 1/2?
Overview
The Riemann Hypothesis states that the nontrivial zeros of the Riemann zeta function ζ(s) have real part exactly 1/2. It connects the distribution of primes to deep properties of complex analysis.
Why It Matters
A proof would sharpen results in analytic number theory and prime distribution. Many theorems are known to follow from RH, and many partial results support it.
Current Status
RH remains unproven. Extensive numerical verification supports the hypothesis, and many conditional results assume RH to obtain stronger bounds.
What a Solution Would Require
A rigorous proof (or disproof) that all nontrivial zeros satisfy Re(s) = 1/2, within accepted analytic number theory standards.