Solve the hardest math in the world

The concept of "the hardest math in the world" is subjective, as difficulty depends on one's background and the current state of mathematical research. However, one of the most famous and challenging unsolved problems is the Riemann Hypothesis.

The Riemann Hypothesis concerns the distribution of prime numbers. It is a conjecture about the zeros of the Riemann zeta function, $\zeta(s)$, which is a function of a complex variable $s = \sigma + it$.

The Riemann zeta function is defined for complex numbers $s$ with real part $\sigma > 1$ by the infinite series:

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \dots$$

This function can be extended to all complex numbers $s \neq 1$ by analytic continuation. The trivial zeros of the zeta function are at $s = -2, -4, -6, \dots$. These are called "trivial" because their existence is relatively easy to prove.

The Riemann Hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part of $\frac{1}{2}$. That is, if $\zeta(s) = 0$ and $s$ is not a trivial zero, then $s$ must be of the form $s = \frac{1}{2} + it$ for some real number $t$.

This hypothesis, proposed by Bernhard Riemann in 1859, is one of the seven Millennium Prize Problems, with a $1 million prize offered for its solution. Its truth would have profound implications for number theory, particularly regarding the distribution of prime numbers. Despite extensive research and computational verification for trillions of zeros, it remains unproven.

The Riemann Hypothesis (unsolved)