The Riemann zeta function or Euler–Riemann zeta function , denoted by
the Greek letter ζ (zeta), is a mathematical function of a complex variable
defined as ζ ( s ) = ∑ n = 1 ∞ 1 n s = 1 1 s + 1 2 s + 1 3 s + ⋯
{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac
{1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots } for Re ( s ) > 1
{\displaystyle \operatorname {Re} (s)>1} , and its analytic continuation
elsewhere.
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 over the reals in the
first half of the eighteenth century.