Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a mathematical function of a complex variable s, and can be expressed as:




ζ
(
s
)
=

∑

n
=
1


∞



n

−
s


=


1

1

s




+


1

2

s




+


1

3

s




+
⋯


{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }
, if



Re
⁡
(
s
)
>
1


{\displaystyle \operatorname {Re} (s)>1}
.The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
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 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. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.

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