Hilbert's problems in the context of "Riemann hypothesis"

David Hilbert (/ˈhɪlbərt/; German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.

Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). He adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century.

A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the Solar System, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox.

Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.

For example, the Diophantine equation ${\displaystyle 3x^{2}-2xy-y^{2}z-7=0}$ has an integer solution: ${\displaystyle x=1,\ y=2,\ z=-2}$. By contrast, the Diophantine equation ${\displaystyle x^{2}+y^{2}+1=0}$ has no such solution.

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. The twin prime conjecture and Goldbach conjecture being special cases of linear diophantine equations became two of four Landau problems.