Hilbert's eighth problem

Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns various branches of number theory, and is actually set of three different problems:

• original Riemann hypothesis for Riemann zeta function
• solvability of two-variable, linear, diophantine equation in prime numbers (where twin prime conjecture and Goldbach conjecture are special cases of this equation)
• generalize methods using Riemann zeta function used to estimate distribution of primes in integers to Dedekind zeta functions and use them for distribution prime ideals in ring of integers of arbitrary number field.

Along with Hilbert's sixteenth problem it become one of the hardest problems on the list, with very few particular results towards its solution. After a century Riemann hypothesis was listed as one of the Smale's problems and Millennium Prize Problems. Twin prime conjecture and Goldbach conjecture being the special cases of linear diophantine equation become two of four Landau problems.

Original statement

Riemann Hypothesis

Essential progress in the theory of the distribution of prime numbers has lately been made by Hadamard, de la Vallée-Poussin, Von Mangoldt and others. For the complete solution, however, of the problems set us by Riemann's paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse," it still remains to prove the correctness of an exceedingly important statement of Riemann, viz., that the zero points of the function zeta(s) defined by the series:

$\zeta (s)=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\dots$

all have the real part 1/2, except the well-known negative integral real zeros. As soon as this proof has been successfully established, the next problem would consist in testing more exactly Riemann's infinite series for the number of primes below a given number and, especially, to decide whether the difference between the number of primes below a number x and the integral logarithm of x does in fact become infinite of an order not greater than 1/2 in x. Further, we should determine whether the occasional condensation of prime numbers which has been noticed in counting primes is really due to those terms of Riemann's formula which depend upon the first complex zeros of the function ${\textstyle \zeta (s)}$.

Linear diophantine equation

After an exhaustive discussion of Riemann's prime number formula, perhaps we may sometime be in a position to attempt the rigorous solution of Goldbach's problem, viz., whether every integer is expressible as the sum of two positive prime numbers; and further to attack the well-known question, whether there are an infinite number of pairs of prime numbers with the difference 2, or even the more general problem, whether the linear diophantine equation:

$ax+by+c=0$

(with given integral coefficients each prime to the others) is always solvable in prime numbers x and y.

Dedekind zeta functions

But the following problem seems to me of no less interest and perhaps of still wider range: To apply the results obtained for the distribution of rational prime numbers to the theory of the distribution of ideal primes in a given number-field $K$ - a problem which looks toward the study of the function ${\textstyle \zeta _{K}(s)}$ belonging to the field and defined by the series:

$\zeta _{K}(s)=\sum {\frac {1}{n(j)^{s}}}$

where the sum extends over all ideals j of the given realm K, and n(j) denotes the norm of the ideal j.

Progress towards solution

In a century after statement of problem by Hilbert many equivalents of Riemann hypothesis was proposed, giving much deeper picture of its significance and more possible ways to prove it. Despite this problem is unsolved.

Despite some much weaker results like Chen's theorem or announced proof of Goldbach's weak conjecture by Harald Helfgott, even special cases of original problem, like twin number conjecture or Goldbach conjecture remains unsolved. General case of diophantine equation given by Hilbert seems to be unable to attack by present tools in number theory.

For Dedekind zeta functions problem was partially resolved: analytic continuation was proven by Erich Hecke along with functional equation. This allows to obtain similar results for prime ideals in rings of integers as current results for distribution of regular prime numbers. However, Extended Riemann hypothesis and much stronger results following from it are still open problem.

References

• Bombieri, Enrico (2006), "The Riemann Hypothesis", The Millennium Prize Problems, Clay Mathematics Institute Cambridge, MA: 107–124
• Hecke, Erich (1983). Mathematische Werke (3 ed.). Göttingen: Vandenhoeck & Ruprecht. pp. 178–197. MR 0749754.

External links

• English translation of Hilbert's original address