Riemann Hypothesis

Hypothesis
All the nontrivial zeroes of the analytic continuation of the Riemann zeta function have a real part equal to $\dfrac 1 2$.

Trivial zeroes occur at every negative even integer ($-2, -4, -6$ etc.).

Critical Line
The line defined by the equation $z = \dfrac 1 2 + i y$ is known as the critical line.

Hence the popular form of the statement of the Riemann hypothesis:
 * "All the nontrivial zeroes of the Riemann zeta function lie on the critical line."

This problem is the first part of no. 8 in the Hilbert 23, and also one of the Millennium Problems, the only one to be in both lists.