View source for Riemann-Siegel formula

'''Lemma 1''' For any complex number <math>z</math>, one has
:<math> \int_{0 \nearrow 1} \frac{e^{i\pi u^2 + 2\pi i z u}}{e^{i\pi u} - e^{-i\pi u}}\ du = \frac{e^{i\pi z} - e^{-i\pi z^2}}{e^{i \pi z} - e^{-i\pi z}}</math>
where <math>0 \nearrow 1</math> denotes a line passing through the line segment <math>[0,1]</math> oriented in the direction <math>e^{i\pi/4}</math>.

'''Proof''' ... <math>\Box</math>

We can rearrange the above lemma as
:<math> \frac{e^{i\pi z}}{e^{i\pi z} -e^{-i\pi z}} = \int_{0 \nearrow 1} \frac{e^{i\pi u^2  + 2\pi i z u}}{e^{i\pi u} - e^{-i\pi u}}\ du + \frac{e^{-i\pi z^2}}{e^{i\pi z} - e^{-i\pi z}}.</math>

Now let <math>s</math> be a complex number with <math>\mathrm{Re} s > 1</math>.  Multiplying both sides of the above equation by <math>(1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) z^{s-1}<math> and integrating on the ray <math>\nwarrow 0</math> from <math>0</math> in the direction <math>e^{3\pi i/4}</math>, we have
:<math> A = B + C</math>
where
:<math>A := (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\nwarrow 0} \frac{z^{s-1} e^{i\pi z}}{e^{i\pi z} -e^{-i\pi z}}\ dz</math>
:<math>B := (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\nwarrow 0} \int_{0 \nearrow 1} \frac{e^{i\pi u^2 + 2\pi i z u} z^{s-1}}{e^{i\pi u} - e^{-i\pi u}}\ du dz</math>
:<math>C := (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\nwarrow 0} \frac{z^{s-1} e^{-i\pi z^2}}{e^{i\pi z} -e^{-i\pi z}}\ dz.</math>

'''Lemma 2''' For any <math>u</math> to the right of the line <math>e^{i\pi/4} {\bf R}</math>, We have
:<math> (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\nwarrow 0} z^{s-1} e^{2\pi i z u}\ du = \pi^{-s/2} \Gamma(s/2) u^{-s}.</math>

'''Proof''' ... <math>\Box</math>

From this Lemma and Fubini (carefully verifying the absolute integrability) we have
:<math>B = \pi^{-s/2} \Gamma(s/2) \int_{0 \nearrow 1} \frac{u^{-s} e^{i\pi u^2}}{e^{i\pi u} - e^{-i\pi u}}\ du.</math>
Similarly, using the geometric series formula
:<math> \frac{e^{i\pi z}}{e^{i\pi z}-e^{-i\pi z}} = -\sum_{n=1}^\infty e^{2\pi i n z}</math>
and Fubini again one has
:<math>A = -\pi^{-s/2} \Gamma(s/2) \zeta(s).</math>
Finally by reflecting the ray <math>\nwarrow 0</math> around the origin and then shifting slightly to the right we have
:<math>C = \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{0 \nwarrow 1} \frac{z^{s-1} e^{-i\pi z^2}}{e^{i\pi z} -e^{-i\pi z}}\ dz,</math>
where <math>0 \nwarrow 1</math> is a line in the direction <math>e^{3\pi i 4}</math> passing through <math>[0,1]</math>.  By analytic continuation we conclude the Riemann-Siegel formula
:<math> \pi^{-s/2} \Gamma(s/2) \zeta(s) = - \pi^{-s/2} \Gamma(s/2) \int_{0 \nearrow 1} \frac{u^{-s} e^{i\pi u^2}}{e^{i\pi u} - e^{-i\pi u}}\ du - 
\pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{0 \nwarrow 1} \frac{z^{s-1} e^{-i\pi z^2}}{e^{-i\pi z} -e^{i\pi z}}\ dz.</math>