What is Riemann hypothesis?

The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics. It was proposed by the German mathematician Bernhard Riemann in 1859. The hypothesis concerns the distribution of non-trivial zeros of the Riemann zeta function, which is a complex function defined for complex numbers \( s = \sigma + it \), where \( \sigma \) and \( t \) are real numbers. The zeta function is initially defined for \( \text{Re}(s) > 1 \) by the infinite 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 Riemann Hypothesis states that all non-trivial zeros of the zeta function, those that lie in the critical strip where \( 0 < \text{Re}(s) < 1 \), have a real part equal to \( \frac{1}{2} \). In other words, if \( s = \sigma + it \) is a non-trivial zero of the zeta function, then: \[ \sigma = \frac{1}{2} \] This conjecture has significant implications for number theory, particularly concerning the distribution of prime numbers. The Riemann 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 the efforts of mathematicians over more than a century, the hypothesis remains unproven.