# Riemann hypotheses

From Encyclopedia of Mathematics

*in analytic number theory*

Five conjectures, formulated by B. Riemann (1876), concerning the distribution of the non-trivial zeros of the 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.

#### References

[a1] | A. Ivic, "The Riemann zeta-function" , Wiley (1985) |

[a2] | E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951) |

[a3] | H.M. Edwards, "Riemann's zeta function" , Acad. Press (1974) pp. Chapt. 3 |

**How to Cite This Entry:**

Riemann hypotheses.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Riemann_hypotheses&oldid=33854

This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article