RELATIONS OF THE DERIVATIVES OF THE RIEMANN ZETA FUNCTION

        Let´s prove today the following beautiful  relations of the derivative of the Riemann Zeta function:


\begin{aligned} &\zeta^{\prime}(-2)=-\frac{\zeta(3)}{4 \pi^{2}} \\ &\zeta^{\prime}(-4)=\frac{3 \zeta(5)}{\pi^{4}} \\ &\zeta^{\prime}(-6)=-\frac{45 \zeta(7)}{8 \pi^{6}} \end{aligned}



We showed previously (here) the functional equation of the Hurwitz Zeta function


\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma(s)}{(2 \pi k)^s}\sum_{r=1}^k\cos\left(\frac{ \pi s}{2}}-\frac{2 \pi r h}{k}\right)\zeta\left(s,\frac{r}{k}\right)
(1)


If we set h=k=1 in (1) we obtain


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


Which is the functional equation of the Riemann zeta function. Let´s now differentiate equation (2) w.r. to s to obtain:


-\zeta^\prime(1-s)=-\frac{2\Gamma(s)}{(2 \pi )^s}\cos\left(\frac{ \pi s}{2}}\right)\zeta\left(s\right)\ln(s)+\frac{2}{(2 \pi )^s}\left(\Gamma^\prime(s)\zeta(s)+\Gamma(s)\zeta^\prime(s)\right)\cos\left(\frac{ \pi s}{2}}\right)-\frac{\pi\Gamma(s)\zeta\left(s\right)}{(2 \pi )^s}\sin\left(\frac{ \pi s}{2}}\right)\\(3)


Letting s=2n+1, n a positive integer and noting that


  \cos\left(\frac{ (2n+1)\pi }{2}}\right)=0

and


  \sin\left(\frac{ (2n+1)\pi }{2}}\right)=\cos(n \pi)\sin\left(\frac{\pi}{2}\right)=(-1)^n 


we may obtain

\zeta^\prime(-2n)=(-1)^n\frac{(2n!)\zeta\left(2n+1\right)}{(2 \pi )^{2n}}\\(4)


Now plugging n=1, n=2 \,\,\text{and}\,\, n=3 in (4) we obtain the desired relations:


\begin{aligned} &\zeta^{\prime}(-2)=-\frac{\zeta(3)}{4 \pi^{2}} \\ &\zeta^{\prime}(-4)=\frac{3 \zeta(5)}{\pi^{4}} \\ &\zeta^{\prime}(-6)=-\frac{45 \zeta(7)}{8 \pi^{6}} \end{aligned}

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