The Riemann Hypothesis is one of the most famous and longstanding unsolved problems in mathematics, specifically in number theory and complex analysis. Proposed by mathematician Bernhard Riemann in 1859, it conjectures that all non-trivial zeros of the Riemann zeta function, denoted by ζ(s), have their real parts equal to 1/2.
The Riemann zeta function is defined for complex numbers s = σ + it (where σ and t are real numbers, and i is the imaginary unit) and is given by the series:
\[ ζ(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \quad \text{for } \text{Re}(s) > 1. \]
This function can be analytically continued to other values of s, except for a simple pole at s = 1. The non-trivial zeros of the Riemann zeta function are those zeros that lie in the "critical strip," where 0 < σ < 1. The Riemann Hypothesis states that all non-trivial zeros lie on the "critical line," where σ = 1/2.
The significance of the Riemann Hypothesis lies in its deep implications for the distribution of prime numbers, as it is closely linked to the Prime Number Theorem, which describes the asymptotic distribution of prime numbers. Proving or disproving 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 solution.