Riemann hypothesis

The Riemann hypothesis conjectures that every nontrivial zero of the Riemann zeta function has real part one-half. It remains unproved.

Riemann zeta function: A complex function whose analytic behavior encodes the distribution of prime numbers. Its nontrivial zeros are exactly the objects whose real parts the hypothesis constrains.
Euler product: A representation of the Riemann zeta function as a product over prime numbers, valid when the real part exceeds one. It links the zeta function to primes before analytic continuation reveals its zeros.
Bernhard Riemann: A German mathematician whose work shaped geometry, complex analysis, and number theory. His 1859 memoir introduced the zeta-function framework and stated the hypothesis.
Riemann hypothesis error bound: A conditional estimate showing that the prime-counting function’s error is bounded on the scale of the square root of x, up to logarithmic factors. This is among the conjecture’s best-known consequences for the distribution of primes.
Hilbert–Pólya conjecture: The proposal that zeta zeros correspond to eigenvalues of a self-adjoint operator. Such an operator could explain why the zeros’ imaginary parts behave like a real spectrum.
Complex analysis: The branch of mathematics that studies functions of complex variables. The zeta function’s analytic continuation and complex zeros are central to formulating the conjecture.
Explicit formula: An identity connecting prime-counting functions with the zeros of the Riemann zeta function. It translates zero locations into oscillations in the distribution of primes.
On the Number of Primes Less Than a Given Magnitude: Bernhard Riemann’s 1859 paper on the distribution of prime numbers. This short memoir contains the original statement and consequences of the conjecture.
Dirichlet L-function: A family of complex functions built from Dirichlet characters and generalizing the Riemann zeta function. The generalized Riemann hypothesis extends the same zero-location claim to these functions.
Random matrix theory and zeta zeros: The study of statistical patterns in zeta zeros using models from random matrix theory. Observed spacing patterns offer evidence and structural clues, but do not prove the hypothesis.

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