Riemann’s ζ-function

Georg Friedrich Bernhard Riemann’s ζ-function is defined

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$

This sum is convergent for ${\mathfrak {Re}}(s)>1$ and defined by analytic continuation elsewhere.

It obeys the functional equation, (as proved in Riemann’s original paper,)

$\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\pi ^{-{\frac {1-s}{2}}}\Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)$

which immediately determines its value for ${\mathfrak {Re}}(s)<0$ in terms of the convergent sum, and has simple zeroes at the negative even integers where $\Gamma \left({\frac {s}{2}}\right)$ is undefined.

It is known that all other zeroes lie on the so-called critical strip where ${\mathfrak {Re}}(s)\in [0,1]$, and hypothesized that in fact ${\mathfrak {Re}}(s)={\frac {1}{2}}$ for all zeroes of $\zeta (s)$ in the critical strip.