Rational Point: def

2025-01-15T22:43:08Z

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Georg Friedrich Bernhard Riemann’s '''ζ-function''' is defined

:<math>\zeta(s)=\sum_{n=1}^\infty\frac1{n^s}</math>

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

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

:<math>\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)
</math>

which immediately determines its value for <math>\mathfrak{Re}(s)<0</math> in terms of the convergent sum, and has simple zeroes at the negative even integers where <math>\Gamma\left(\frac s2\right)</math> is undefined.

It is known that all other zeroes lie on the so-called '''''critical strip''''' where <math>\mathfrak{Re}(s)\in[0,1]</math>, and hypothesized that in fact <math>\mathfrak{Re}(s)=\frac12</math> for all zeroes of <math>\zeta(s)</math> in the critical strip.

Riemann’s ζ-function - Revision history

Rational Point: def