Riemann’s ζ-function
Georg Friedrich Bernhard Riemann’s ζ-function is defined
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s)=\sum_{n=1}^\infty\frac1{n^s}}
This sum is convergent for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{Re}(s)>1} and defined by analytic continuation elsewhere.
It obeys the functional equation, (as proved in Riemann’s original paper,)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{-\frac s2}\Gamma\left(\frac s2\right)\zeta(s)= \pi^{-\frac{1-s}2}\Gamma\left(\frac{1-s}2\right)\zeta(1-s) }
which immediately determines its value for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{Re}(s)<0} in terms of the convergent sum, and has simple zeroes at the negative even integers where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma\left(\frac s2\right)} is undefined.
It is known that all other zeroes lie on the so-called critical strip where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{Re}(s)\in[0,1]} , and hypothesized that in fact Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathfrak{Re}(s)=\frac12} for all zeroes of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(s)} in the critical strip.
