Riemann zeta function



The Riemann zeta function or Euler–Riemann zeta function, $ζ(s)$, is a function of a complex variable s that analytically continues the sum of the Dirichlet series


 * $$\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s}$$

which converges when the real part of $s$ is greater than 1.

Using the techniques of analytic continuation we can determine the values when the real part of $s$ is less than 1. For example:


 * ζ(−1) = $− 1⁄12$.

But pluging a value of -1 into the first equation gives


 * $$\zeta(-1) =\sum_{n=1}^\infty {n}$$