Difference between revisions of "Riemann hypotheses"
From Encyclopedia of Mathematics
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''in analytic number theory'' | ''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]] | + | 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 $\operatorname{Re} s = 1/2$. | ||
Revision as of 12:58, 9 December 2012
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://encyclopediaofmath.org/index.php?title=Riemann_hypotheses&oldid=29141
Riemann hypotheses. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_hypotheses&oldid=29141
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article