The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field of number theory. It is concerned with the distribution of prime numbers and is named after the German mathematician Bernhard Riemann, who first formulated the hypothesis in 1859.
At its core, the Riemann Hypothesis posits that all "non-trivial" zeros of the Riemann zeta function \(\zeta(s)\) lie on a critical line in the complex plane, specifically that all such zeros have a real part equal to \(\frac{1}{2}\).
The Riemann zeta function is defined for complex numbers \(s = \sigma + it\) (where \(\sigma\) and \(t\) are real numbers) and is initially defined for \(\text{Re}(s) > 1\) by the series:
\[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \]
It can be analytically continued to other values of \(s\), except for \(s = 1\), where it has a simple pole.
The non-trivial zeros of the zeta function are the values of \(s\) for which \(\zeta(s) = 0\) and are located in the critical strip where \(0 < \text{Re}(s) < 1\). The hypothesis asserts that if \(s\) is a non-trivial zero, then \(s = \frac{1}{2} + it\) for some real number \(t\).
The Riemann Hypothesis has profound implications for number theory, particularly for the distribution of prime numbers, as it is deeply related to the Prime Number Theorem and the understanding of how primes are spaced apart.
The hypothesis is one of the seven "Millennium Prize Problems" for which the Clay Mathematics Institute has offered a prize of one million dollars for a correct proof or counterexample. Despite extensive numerical evidence supporting the hypothesis and its connections to many areas of mathematics, it remains unproven.