The Riemann Hypothesis states that every non-trivial zero of the Riemann zeta function has real part exactly 1/2. Proposed by Bernhard Riemann in 1859, it remains unsolved after more than 160 years and is widely considered the most important open problem in mathematics. It is one of the seven Millennium Prize Problems, carrying a $1 million reward for a correct proof.
Prime numbers (2, 3, 5, 7, 11, 13, ...) look scattered and unpredictable — there's no simple formula that tells you the next one. But mathematicians have found a deep, hidden structure connecting primes to a special function, and one unresolved question about that function is considered the greatest open problem in mathematics: the Riemann Hypothesis.
Bernhard Riemann studied a function (now called the Riemann zeta function) that, for certain special complex numbers called "zeros," equals exactly zero. Some of these zeros are simple and well understood ("trivial zeros"). But there are infinitely many other, more mysterious zeros, and Riemann conjectured that every single one of them shares a very specific property — they all lie on one particular line, when plotted on a special kind of graph. That's the whole hypothesis. It's been checked for the first 10 trillion zeros, and every single one fits — but "checked for 10 trillion cases" is not the same as "proven for all infinitely many cases."
The Riemann Hypothesis has resisted proof for over 160 years despite attempts by some of history's greatest mathematicians. It's one of the 7 "Millennium Prize Problems" selected by the Clay Mathematics Institute in 2000, each carrying a $1 million prize. As of 2026, it remains the only one of the seven still completely unsolved (the Poincaré Conjecture was solved in 2003 by Grigori Perelman, who famously declined the prize money).
For a complex number s with real part greater than 1, the zeta function is defined by the infinite sum:
Riemann showed this function can be extended (via analytic continuation) to almost all complex numbers, except s = 1 (where it has a pole — the function blows up to infinity).
The zeta function equals zero at the negative even integers: −2, −4, −6, .... These are called the trivial zeros — their existence and location are fully understood. All other zeros — the non-trivial zeros — lie in the "critical strip" where the real part of s is between 0 and 1.
Riemann's conjecture: every non-trivial zero has real part exactly 1/2. In other words, if you plot the non-trivial zeros on the complex plane, they all sit on a single vertical line — the "critical line" Re(s) = 1/2.
Euler had earlier shown a remarkable identity connecting the zeta function to primes (the "Euler product"):
This identity is the bridge between the zeta function and prime numbers. Riemann used it to derive an exact formula for the number of primes up to any given number, expressed as a sum involving the zeta function's zeros. The more precisely we know where the zeros are, the more precisely we know how primes are distributed — which is why the Riemann Hypothesis, if true, would give the best possible bound on the error term in the Prime Number Theorem.
The first 10¹³ (10 trillion) non-trivial zeros have been computed and all lie exactly on the critical line, providing overwhelming (but not conclusive) evidence for the hypothesis. Numerical evidence, however extensive, does not constitute a mathematical proof.
Hardy (1914) proved infinitely many zeros lie on the critical line — but this doesn't establish that ALL of them do. Levinson (1974) showed at least 1/3 of the zeros lie on the critical line; Conrey (1989) improved this to over 40%. The Prime Number Theorem itself (Hadamard and de la Vallée-Poussin, 1896) was proven using the weaker fact that ζ(s) has no zeros with real part exactly 1 — a much easier result than the full hypothesis, but it was enough to establish the basic asymptotic law of prime distribution.
Hundreds of mathematical results are currently stated as "true, assuming the Riemann Hypothesis" — a conditional proof strategy widely used in analytic number theory. A proof of RH would immediately validate all of these, including improved bounds on prime gaps, more precise estimates in the Prime Number Theorem, and results in cryptography relying on the difficulty of certain number-theoretic problems.
The Riemann-Siegel formula, developed from unpublished notes of Riemann found by Siegel in 1932, provides an efficient method for computing zeta zeros numerically, enabling the large-scale verification efforts (10¹³ zeros and counting) conducted since the late 20th century using specialized algorithms and distributed computing.
A broader conjecture, the Generalized Riemann Hypothesis (GRH), extends the same critical-line claim to a wider class of functions called Dirichlet L-functions. GRH would have even more far-reaching consequences, including implications for the distribution of primes in arithmetic progressions and for certain primality testing algorithms.
The zeta function's zeros are governed by deep and subtle interactions between analysis (the theory of complex functions) and number theory (the discrete structure of primes) — two areas that don't naturally mesh well. Many proof strategies attempted over the past century (including approaches via random matrix theory, following an unexpected statistical connection discovered by Montgomery and Dyson in the 1970s, and approaches via non-commutative geometry, pursued by Alain Connes) have provided insight but no complete proof. The problem is now approaching two centuries old.