Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase 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 R e ( s ) > 1 {\displaystyle \mathrm {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.