Riemann Siegel function and gamma function

Question

I ask about an idea to prove this formula:

$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$

where $ϑ(β)$ is the Riemann Siegel function.

Answer 1

I know two proof, the first uses $$\cos\frac{\pi s}{2}=\frac{1}{\sqrt{2}}\sqrt{\cosh(\pi t)}\,e^{-i\arctan(\tanh\frac{\pi t}{2})}.\qquad (1)$$ and $$\Gamma(\frac12+i\frac t 2)=|\Gamma(\frac14+i\frac t2)|\,e^{i(\vartheta(t)+\frac t 2\log\pi)},\qquad (2)$$

Since $$\Gamma(z)\Gamma(z+1/2)=2^{1-2z}\sqrt{\pi}\Gamma(2z);\quad \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}$$ we get $$\Gamma(2z)=2^{2z-1}\pi^{-1/2} \Gamma(z)\frac{\pi}{\cos\pi z}\frac{1}{\Gamma(1/2-z)}.$$ We put now $z=\frac14+i\frac t 2$, $t$ real $$ \Gamma(\frac12+it)=2^{-\frac12+it}\frac{\pi^{\frac12}}{\cos\pi(\frac14+i\frac{t}{2})} \frac{\Gamma(\frac14+i\frac{t}{2})}{\Gamma(\frac14-i\frac{t}{2})}. $$ From (1) and (2) we get $$\Gamma(1/2+it)=2^{it}\frac{\pi^{1/2}}{\sqrt{\cosh \pi t}\; e^{-i \arctan\tanh(\pi t/2)}}\pi^{it} e^{2i\vartheta(t)}. $$ so that $$ \Gamma(1/2+it)=\sqrt{\frac{\pi}{\cosh\pi t}}\exp\bigl\{i(2\vartheta(t)+t\log(2\pi)+\arctan\tanh(\pi t/2))\bigr\} $$

The other proof I know uses the functional equation of the zeta function.

Answer 2

I extended juan's proof to a formula for all complex numbers.

Consequently, we get the Euler chi function $\chi(z):=\zeta(1-z)/\zeta(z)$.

(substitution)

$\tan^{-1}(\tanh(\pi t/2))\longrightarrow\frac{\mathrm{gd}(\pi t)}2$, where $\mathrm{gd}(z)$ is the Gudermannian function.

$\sqrt{\frac{\pi}{\cosh(\pi t)}}\longrightarrow\frac{\sqrt{\pi}}{\cosh(\pi t)}\left(\sinh(\frac{\pi t}2)\sin(\frac{\mathrm{gd}(\pi t)}{2})+\cosh(\frac{\pi t}{2})\cos(\frac{\mathrm{gd}(\pi t)}2)\right)$

$t\longrightarrow z$

(simplification) If we covert the above result(trigonometric functions) to exponentials, then we get the following simplified formula.

$\Gamma(\frac12+iz)=\frac{\sqrt{\pi}(1+i)(2\pi)^{iz}}{e^{\pi z}+i}e^{\frac{\pi z}2+2i\vartheta(z)}/;z\in\mathbb{C}$

Therefore,

$\Gamma(z)=\frac{(2\pi)^z}{1+e^{i\pi z}}e^{i(2\vartheta(\frac{i}2-iz)+\frac{\pi z}2)}=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}e^{2i\vartheta(\frac{i}2-iz)}=\frac{(2\pi)^z}{2\cos(\frac{\pi z}2)}\chi(z)$