The Riemann zeta function

The Riemann zeta-function is defined by

$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s} =\prod_p (1-p^{-s})^{-1} ,$$

where $s=\sigma+it$, the product is over the primes, and the series and product converge absolutely for $\sigma>1$.

The use of $s=\sigma+it$ as a complex variable in the theory of the Riemann $\zeta$-function has been standard since Riemann's original paper.

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