The Riemann hypothesis
The Riemann zeta function is defined by for and extended by analytic continuation to all complex . It vanishes at . Prove that every other zero of has real part exactly .
History and references
Still open as of 2026.
Riemann made the conjecture in his 1859 paper on the number of primes below a given bound, where the zeros of zeta govern how regularly the primes are distributed. Together with the Goldbach and twin prime conjectures it formed Hilbert's eighth problem in 1900, and it is one of the Clay Millennium Prize Problems. Vast numbers of zeros have been computed and all lie on the line . Analogues for curves over finite fields were proved by André Weil, but the original statement remains open. Many consider it the most important unsolved problem in pure mathematics.
References
Solutions 0
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