Difference between revisions of "Riemann hypotheses"
From Encyclopedia of Mathematics
(Importing text file) |
(Category:Number theory) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
| + | |||
''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 $\operatorname{Re} s = 1/2$. | ||
| Line 9: | Line 15: | ||
====References==== | ====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> | + | <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> | ||
| + | |||
| + | [[Category:Number theory]] | ||
Latest revision as of 20:17, 18 October 2014
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=13088
Riemann hypotheses. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_hypotheses&oldid=13088
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article