the Riemann hypothesis
/ REE-mahn /
The Riemann hypothesis is the most famous unsolved problem in mathematics. Stated as briefly as possible: every non-trivial zero of the Riemann zeta function has real part exactly 1/2. That is, apart from the well-understood trivial zeros at s = -2, -4, -6, ..., every place where zeta(s) = 0 lies precisely on the critical line Re s = 1/2. Riemann conjectured it in his 1859 paper, and despite a century and a half of effort and overwhelming numerical evidence, no one has proved or disproved it.
Why would anyone believe such a precise claim, and why care? Because zeta encodes the primes through its Euler product, and the LOCATION of its zeros translates directly into the regularity of the primes. There is an explicit formula expressing the prime counting function as a 'main term' plus a sum of oscillating contributions, one per non-trivial zero, where each zero at s = beta + i gamma contributes an error of size roughly x^beta. If all beta = 1/2, every error term is as small as it can possibly be — the primes are then distributed as evenly as the prime number theorem allows. The Riemann hypothesis is, in this sense, the assertion that there is no hidden conspiracy among the primes, no zero lurking off the line to create unexpected clumping.
The evidence is staggering — the first many trillions of zeros have been computed and all lie exactly on the line, a positive proportion is proven to lie on it, and countless theorems are known 'assuming RH'. But evidence is not proof, and the history of number theory is full of patterns that hold for astronomically long before failing. Be honest about the status: RH is OPEN. It is one of the seven Clay Millennium Prize Problems, generalises to a vast web of L-functions (the Generalised Riemann Hypothesis), and a proof or disproof would reshape analytic number theory. Anyone claiming an elementary proof should be met with deep scepticism.
If RH holds, the prime counting function pi(x) (the number of primes up to x) satisfies pi(x) = Li(x) + O(sqrt(x) log x), an error bound that is essentially optimal. Without RH the best unconditional error is far weaker. A single off-line zero at, say, real part 0.6 would inflate the error to about x^0.6 — visibly larger fluctuations in how the primes thin out.
RH is exactly the statement that the prime counting error is as small as possible — every non-trivial zero on the line Re s = 1/2.
RH is unproven. Treat 'assuming the Riemann hypothesis' as a hypothesis, not a fact; many published theorems are conditional on it. Numerical verification of trillions of zeros is strong evidence but logically proves nothing about the infinitely many zeros not yet checked.