View source for Riemann hypotheses

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''in analytic number theory''

Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the [[Zeta-function|zeta-function]]
\begin{equation}
\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},\quad s=\sigma+it,
\end{equation}
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 $\operatorname{Re} s = 1/2$.



====Comments====
For the list of all 5 conjectures see [[Zeta-function|Zeta-function]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,   "The Riemann zeta-function" , Wiley  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.C. Titchmarsh,   "The theory of the Riemann zeta-function" , Clarendon Press  (1951)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.M. Edwards,   "Riemann's zeta function" , Acad. Press  (1974)  pp. Chapt. 3</TD></TR></table>