Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, and in particular the Riemann hypothesis, although it is also concerned with the

Goldbach Conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ...

. The problem as stated asked for more work on the distribution of primes and generalizations of Riemann hypothesis to other rings where

prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...

s take the place of primes. Riemann zeta function absolute value

Subtopics

Riemann hypothesis and generalizations

Hilbert calls for a solution to the Riemann hypothesis, which has long been regarded as the deepest open problem in mathematics. Given the solution, he calls for more thorough investigation into Riemann's

zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * ...

and the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...

Goldbach conjecture

He 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.

Twin prime conjecture

Generalized Riemann conjecture

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

External links

English translation of Hilbert's original address
#08

References

* * {{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, url=https://www.sciencedirect.com/science/article/pii/S2666657X2100001X