Collection: Riemann Zeta Function

The Riemann zeta function is related to the distribution of prime numbers. It also has uses in other areas such as physics, probability theory, and applied statistics. It is named after the German mathematician Bernhard Riemann, who wrote about it in the memoir "On the Number of Primes Less Than a Given Quantity", published in 1859.

Leonhard Euler first introduced and studied the function in the first half of the eighteenth century, using only real numbers, as complex analysis was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.

The Riemann zeta function is related to one of the greatest unsolved problems in mathematics: the Riemann hypothesis, classified as the 8th of Hilbert's 23 problems.

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function. Many mathematicians consider the Riemann hypothesis to be the most important unsolved problem in pure mathematics. [1] In the year 2000, the Clay Mathematics Institute included the Riemann hypothesis as one of their Millennium Prize Problems, promising a reward of US$1 million to anyone who could solve it.