Riemann-Siegel formula

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Lemma 1 For any complex number [math]z[/math], one has

[math] \int_{0 \uprightarrow 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 \uprightarrow 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 \uprightarrow 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 \gt 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}\ltmath\gt and integrating on the ray \ltmath\gt\upleftarrow 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_{\upleftarrow 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_{\upleftarrow 0} \int_{0 \uprightarrow 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_{\upleftarrow 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} \R\ltmath\gt, We have :\ltmath\gt (1 + e^{-i\pi s) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\upleftarrow 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 \uprightarrow 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]\upleftarrow 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 \upleftarrow 1} \frac{z^{s-1} e^{-i\pi z^2}}{e^{i\pi z} -e^{-i\pi z}}\ dz,[/math]

where [math]0 \upleftarrow 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 \uprightarrow 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 \upleftarrow 1} \frac{z^{s-1} e^{-i\pi z^2}}{e^{-i\pi z} -e^{i\pi z}}\ dz.[/math]