# Riemann hypotheses

in analytic number theory

Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the zeta-function $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},\quad s=\sigma+it,$$ and the expression via these zeros of the number of prime numbers not exceeding a real number $x$. One of the Riemann hypotheses has neither been proved nor disproved: All non-trivial zeros of the zeta-function $\zeta(s)$ lie on the straight line $\operatorname{Re} s = 1/2$.