Riemann hypothesis

The Riemann hypothesis is the conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2, so they lie on the critical line. The article says it is one of the most important unsolved problems in pure mathematics and that there is strong numerical evidence but no proof.

It is tied to the distribution of prime numbers through Riemann’s explicit formula. The text also says it has many equivalent statements and consequences, including bounds for prime-counting errors, growth of arithmetic functions, and analogues for other zeta and L-functions.

Reading notes

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• Conjecture that all non-trivial zeros of the Riemann zeta function have real part equal to 1/2.
• The trivial zeros occur at the negative even integers.
• It is described as one of the most important open problems in pure mathematics.
• It is of interest to number theory because it implies results about the distribution of prime numbers.
• The statement was proposed by Bernhard Riemann.
• There is strong numerical evidence in favor, but no known proof.
• The hypothesis is equivalent to several statements about the zeta function, arithmetic functions, and the Farey sequence.
• The article links the hypothesis to bounds for the error in the prime-counting function and to the Mertens function.
• It also relates the hypothesis to analytic criteria such as Riesz, Nyman-Beurling, Salem, Weil, Li, and Speiser.
• The text records consequences for gaps between primes and for the behavior of functions such as μ, σ, and φ.
• The hypothesis has generalizations for Dirichlet L-functions, zeta functions of number fields, and other contexts.
• There are proven analogies in finite fields, curves, graphs, Selberg functions, and Ihara functions.
• The behavior of zeros on the critical line is compared to random Hermitian matrices and to ideas of quantum chaos.
• The article also brings together numerical checks and partial results about zeros on the critical line.