# Riemann-Siegel formula

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Lemma 1 For any complex number $z$, one has

$\int_{0 \uprightarrow 1} \frac{e^{i\pi u^2} + 2\pi i z u}}{e^{i\pi u} - e^{-i\pi u}}\ du = \frac{e^{i\pi z} - e^{-i\pi z^2}}{e^{i \pi z} - e^{-i\pi z}}$

where $0 \uprightarrow 1$ denotes a line passing through the line segment $[0,1]$ oriented in the direction $e^{i\pi/4}$.

Proof ... $\Box$

We can rearrange the above lemma as

$\frac{e^{i\pi z}}{e^{i\pi z} -e^{-i\pi z}} = \int_{0 \uprightarrow 1} \frac{e^{i\pi u^2} + 2\pi i z u}}{e^{i\pi u} - e^{-i\pi u}}\ du + \frac{e^{-i\pi z^2}}{e^{i\pi z} - e^{-i\pi z}}.$

Now let $s$ be a complex number with $\mathrm{Re} s \gt 1$. Multiplying both sides of the above equation by $(1 + e^{-i\pi s) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) z^{s-1}\ltmath\gt and integrating on the ray \ltmath\gt\upleftarrow 0$ from $0$ in the direction $e^{3\pi i/4}$, we have

$A = B + C$

where

$A := (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\upleftarrow 0} \frac{z^{s-1} e^{i\pi z}}{e^{i\pi z} -e^{-i\pi z}}\ dz$
$B := (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\upleftarrow 0} \int_{0 \uprightarrow 1} \frac{e^{i\pi u^2} + 2\pi i z u} z^{s-1}}{e^{i\pi u} - e^{-i\pi u}}\ du dz$
$C := (1 + e^{-i\pi s}) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\upleftarrow 0} \frac{z^{s-1} e^{-i\pi z^2}}{e^{i\pi z} -e^{-i\pi z}}\ dz.$

Lemma 2 For any $u$ to the right of the line $e^{i\pi/4} \R\ltmath\gt, We have :\ltmath\gt (1 + e^{-i\pi s) \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{\upleftarrow 0} z^{s-1} e^{2\pi i z u}\ du = \pi^{-s/2} \Gamma(s/2) u^{-s}.$

Proof ... $\Box$

From this Lemma and Fubini (carefully verifying the absolute integrability) we have

$B = \pi^{-s/2} \Gamma(s/2) \int_{0 \uprightarrow 1} \frac{u^{-s} e^{i\pi u^2}}{e^{i\pi u} - e^{-i\pi u}}\ du.$

Similarly, using the geometric series formula

$\frac{e^{i\pi z}}{e^{i\pi z}-e^{-i\pi z}} = -\sum_{n=1}^\infty e^{2\pi i n z}$

and Fubini again one has

$A = -\pi^{-s/2} \Gamma(s/2) \zeta(s).$

Finally by reflecting the ray $\upleftarrow 0$ around the origin and then shifting slightly to the right we have

$C = \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{0 \upleftarrow 1} \frac{z^{s-1} e^{-i\pi z^2}}{e^{i\pi z} -e^{-i\pi z}}\ dz,$

where $0 \upleftarrow 1$ is a line in the direction $e^{3\pi i 4}$ passing through $[0,1]$. By analytic continuation we conclude the Riemann-Siegel formula

$\pi^{-s/2} \Gamma(s/2) \zeta(s) = - \pi^{-s/2} \Gamma(s/2) \int_{0 \uprightarrow 1} \frac{u^{-s} e^{i\pi u^2}}{e^{i\pi u} - e^{-i\pi u}}\ du - \pi^{-\frac{1-s}{2}} \Gamma(\frac{1-s}{2}) \int_{0 \upleftarrow 1} \frac{z^{s-1} e^{-i\pi z^2}}{e^{-i\pi z} -e^{i\pi z}}\ dz.$