Let $n$ be a positive real number. Can the equality

$$\dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\Big(\pi^{-s/2}\Gamma\Big(1+\frac{s}{2}\Big)\Big)\Big]\Bigg|_{s=1} = - \dfrac{d^{n}}{ds^{n}}\Big[s^{n-1}\ln\Big(\ln(s-1)\zeta(s)\Big)\Big]\Bigg|_{s=1}$$

be possible for any positive real $n$, where $\zeta(s)$ is the Riemann zeta function and $\Gamma(s)$ is the usual gamma function in number theory ?

My approach was by fractional calculus (to accommodate all reals), but I did not complete it since it appeared terribly malicious too me. I'm wondering if there can be some shorter and more intuitive way? Even a long complete proof by fractional calculus will still be very much appreciated.

extramessy ? $\endgroup$9more comments