Editing Hilbert's eighth problem

{{Short description|On the distribution of prime numbers}}
'''Hilbert's eighth problem''' is one of [[David Hilbert]]'s [[Hilbert's problems|list of open mathematical problems]] posed in 1900. It concerns [[number theory]], and in particular the [[Riemann hypothesis]],{{sfnp|Bombieri|2006}} although it is also concerned with the [[Goldbach conjecture]]. It asks for more work on the [[distribution of primes]] and generalizations of Riemann hypothesis to other [[ring (mathematics)|ring]]s where [[prime ideal]]s take the place of primes.

[[File:Riemann zeta function absolute value.png|thumb|250px|Absolute value of the ζ-function. Hilbert's eighth problem includes the [[Riemann hypothesis]], which states that this function can only have non-trivial zeroes along the line&nbsp;''x''&nbsp;=&nbsp;1/2 {{sfnp|Moxley|2021}}.]]

==Riemann hypothesis and generalizations==
{{Main|Riemann hypothesis}}

Hilbert calls for a solution to the Riemann hypothesis, which has long been regarded as the deepest open problem in mathematics. Given the solution,{{sfnp|Moxley|2021}} he calls for more thorough investigation into Riemann's [[Riemann zeta function|zeta function]] and the [[prime number theorem]].

==Goldbach conjecture==
{{Main|Goldbach conjecture}}

Hilbert calls for a solution to the Goldbach conjecture, as well as more general problems, such as finding infinitely many pairs of primes solving a fixed linear [[diophantine equation]].

==Generalized Riemann conjecture==
{{Main|Generalized Riemann hypothesis}}

Finally, Hilbert calls for mathematicians to generalize the ideas of the Riemann hypothesis to counting prime ideals in a number field.

== External links ==
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob8 English translation of Hilbert's original address]

{{Hilbert's problems}}

[[Category:Hilbert's problems|#08]]

== References ==

* {{citation|last=Bombieri|first= Enrico|title= The Riemann Hypothesis|journal=The Millennium Prize Problems |volume=Clay Mathematics Institute Cambridge, MA|pages= 107–124 |year=2006|url=https://bookstore.ams.org/mprize}}
* {{citation|last=Moxley|first= Frederick|title= Complete solutions of inverse quantum orthogonal equivalence classes|journal=Examples and Counterexamples |volume=1|pages= 100003 |year=2021|doi= 10.1016/j.exco.2021.100003|doi-access=free}}