(number theory, analytic number theory, uncountable) The function ζ defined by the Dirichlet series \textstyle \zeta(s)=\sum_{n=1}^\infty \frac 1 {n^s} = \frac1{1^s} + \frac1{2^s} + \frac1{3^s} + \frac1{4^s} + \cdots, which is summable for points s in the complex half-plane with real part > 1; the analytic continuation of said function, being a holomorphic function defined on the complex numbers with pole at 1.