# zeta function-deriving reflection equation from functional equation

From analytic continuation of zeta function,

\begin{align} \zeta(s) = \frac{\pi^{\frac{s}{2}}}{\Gamma\left(\frac{s}{2}\right)} \left[ \frac{1}{s(s-1)} + \int_1^{\infty} \left( x^{\frac{s}{2}-1} + x^{-\frac{s}{2}-1} \right) \left(\frac{\theta(x) - 1}{2}\right) dx \right] \end{align}

From this I obtain a functional equation \begin{align} \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s) = \pi^{-\frac{1-s}{2}} \Gamma\left(\frac{1-s}{2}\right) \zeta(1-s) \end{align} My next step is to prove reflection functional equation

\begin{align} \zeta(1-s) = 2 (2\pi)^{-s} \cos\left(\frac{s\pi}{2}\right) \Gamma(s)\zeta(s) \end{align}

Reordering a functional equation for $$\zeta(s)$$, I have \begin{align} \zeta(1-s) = \pi^{\frac{1}{2}} \frac{\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{1-s}{2}\right)} \zeta(s) \end{align} I have problem with showing above two equations are indeed same...

Let $$F(s) = \Gamma\left(1-\frac{s}{2}\right) \left( \Gamma\left(\frac{1-s}{2}\right) 2 (2\pi)^{-s}\Gamma(s) -\frac{\pi^{\frac{1}{2}}\Gamma\left(\frac{s}{2}\right)}{ \cos\left(\frac{s\pi}{2}\right)}\right)$$ From $$\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}dt$$,$$\Gamma(s+1) = s \Gamma(s), \Gamma(1) = 1$$ we obtain that $$F$$ is $$4$$-periodic.
From $$\Gamma(1/2) = \sqrt{\pi}$$ we find the poles cancel each other so that $$F$$ is entire.
$$F$$ is bounded because $$\Gamma(s+1)$$ is the Fourier transform of $$e^u e^{-e^{u}}$$ whose all derivatives are $$L^1$$, thus $$\Gamma(s+1)$$ is rapidly decreasing on vertical lines $$\Re(s ) > 0$$ and $$\Gamma(s+1) = s \Gamma(s)$$ implies it is rapidly decreasing on all vertical lines.
That is $$F$$ is a bounded entire function, thus it is constant, $$F(s) = F(i\infty) = 0$$.
Use \begin{align} &\frac{\pi^{\frac{1}{2}}}{2^{s-1}} \Gamma (s) = \Gamma \left(\frac{s}{2}\right) \Gamma \left(\frac{s+1}{2}\right) \\ & \Gamma\left(\frac{1+s}{2}\right) \Gamma\left(\frac{1-s}{2}\right) = \frac{\pi}{\cos\left(\frac{\pi s}{2}\right)} \end{align} which comes from Legndre duplication formula and Euler's Reflection formula of gamma function.