Riemann zeta function

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The Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series

which converges when the real part of s is greater than 1. It plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.

First results about this function were obtained by Leonhard Euler in the eighteenth century. It is named after Bernhard Riemann, who in the memoir "On the Number of Primes Less Than a Given Magnitude", published in 1859, established a relation between its zeros and the distribution of prime numbers.[1]

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 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. Many generalizations of the Riemann zeta function, such as Dirichlet series and L-functions, are known.


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