The function $\zeta \left(s\right)={\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^s}}$ defined on the complex variables s. The sum is convergent, and gives an analytic function, when Res > 1, and in particular $\zeta \left(2\right)=\frac{{\pi }^2}{6};\zeta \left(4\right)=\frac{{\pi }^4}{90};\zeta \left(6\right)=\frac{{\pi }^6}{945}.$ This definition can then be analytically extended to any s ≠ 1. The zero function has ‘trivial’ zeros at s = –2, –4, –6,… and all other zeros are conjectured to be on the ‘critical line’ Res = ½ (see Riemann hypothesis).

There are deep connections between the zeta function and the exact distribution of the primes, as Riemann found an expression for this distribution in terms of the zeros of the zeta function.