RiemannHypothese

The Riemann Hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function. The Riemann zeta function, denoted by the Greek letter zeta (ζ), is a function of a complex variable s, defined for Re(s) > 1 by the infinite series ζ(s) = 1/1^s + 1/2^s + 1/3^s + ..., and by analytic continuation elsewhere in the complex plane. The hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2. The trivial zeros are the negative even integers (-2, -4, -6, ...).

The non-trivial zeros are the complex numbers s for which ζ(s) = 0, excluding the trivial zeros. Bernhard Riemann first proposed this hypothesis in 1859. Proving or disproving the Riemann Hypothesis would have profound implications for number theory, particularly for the distribution of prime numbers. Many theorems in analytic number theory are proven assuming the truth of the Riemann Hypothesis, and are thus conditional.

The distribution of prime numbers is closely related to the zeros of the zeta function. If the hypothesis is true, it would provide a very precise understanding of how primes are distributed among the integers. Despite extensive computational verification, with trillions of zeros found to lie on the critical line (where the real part is 1/2), no mathematical proof has yet been established. It remains one of the most important unsolved problems in mathematics, and is one of the seven Millennium Prize Problems.