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 \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)

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[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

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From The Encyclopedia of Math: Riemann hypotheses.

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