Theta Function - Relation To The Riemann Zeta Function

Relation To The Riemann Zeta Function

The relation

was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the integral

$\Gamma\left(\frac{s}{2}\right) \pi^{-s/2} \zeta(s) = \frac{1}{2}\int_0^\infty\left t^{s/2}\frac{dt}{t}$

which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z not zero is given in the article on the Hurwitz zeta function.

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