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 a set of three different problems:

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

Along with Hilbert's sixteenth problem, it became one of the hardest problems on the list, with very few particular results towards its solution. After a century, the Riemann hypothesis was listed as one of Smale's problems and the Millennium Prize Problems. ^[1] The twin prime conjecture and Goldbach conjecture being special cases of linear diophantine equations became 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:

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

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:

${\displaystyle ax+by+c=0}$

(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 ${\displaystyle K}$ - a problem which looks toward the study of the function ${\textstyle \zeta _{K}(s)}$ belonging to the field and defined by the series:

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

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 the statement of the problem by Hilbert, many equivalents of the Riemann hypothesis were proposed, giving a much deeper picture of its significance and more possible ways to prove it.

The general case of diophantine equations given by Hilbert seems to be unable to attack using present tools in number theory. Results about bounds of gaps between primes obtained by Yitang Zhang and later improved by Polymath project provides infinite number of prime solutions for very special case:

${\displaystyle x-y+c=0{\text{, when: }}c\geq 246}$

Some much weaker results following from twin prime conjecture and Goldbach conjecture, like Chen's theorem or the attempted proof of Goldbach's weak conjecture by Harald Helfgott (currently under review), gives reason to believe in truth of original conjectures, but don't provide answers about prime solutions of equation given by Hilbert.

For Dedekind zeta functions, the status of the problem depends what kind of results one expects from Dedekind zeta functions. Existence of analytic continuation for them was proven by Erich Hecke along with functional equation. ^[2] This allowed to obtain similar results for prime ideals in rings of integers as for usual primes with current knowledge about Riemann zeta function. However, if one interpret this in context of the first point of the problem as a challenge to prove extended Riemann hypothesis, and thus obtain the much stronger results following from it, this part of the problem is still unresolved.

References

1. Bombieri (2006).
2. Hecke (1983).

• 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