What function satisfies the following equation?

Question

$$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$

I think it should be similar to Zeta function, but what is it exactly?

Answer 1

You are right about the similarity of $f(x)$ to the Riemann Zeta function.

Using the following identity that can be found here $$\pi^{-\frac{z}{2}}\Gamma\left(\frac{z}{2}\right)\zeta(z)=\pi^{-\frac{1-z}{2}}\Gamma\left(\frac{1-z}{2}\right)\zeta(1-z)$$ and setting $z=\frac{2x}{\pi}$ results in $$\color{red}{\pi^{-\frac{x}{\pi}}}\Gamma\left(\frac{x}{\pi}\right)\color{red}{\zeta\left(\frac{2x}{\pi}\right)}=\color{red}{\pi^{-\frac{1}{2}-\frac{x}{\pi}}}\Gamma\left(\frac{1}{2}-\frac{x}{\pi}\right)\color{red}{\zeta\left(1-\frac{2x}{\pi}\right)}$$ Comparing this expression (paying particular attention to the parts highlighted in red) with the relationship given in the question, we can deduce that $$f(x)=\pi^{-\frac{x}{\pi}}\zeta\left(\frac{2x}{\pi}\right)e^{x}$$