Summation of gamma function in terms of reimann zeta function

Question

reimann zeta function is new for me.Can anyone tell how is it related to gamma function by summation & how author got 2nd line after solving summation in first line

i thought using following expression but it didn't work out

$\zeta(1-s)=2^{1-s}\pi^{-s}\cos\left(\dfrac{s\pi}{2}\right)\zeta(s)\Gamma(s)$

Answer 1

We do not need to use the functional equation here.

He got to the second line by using the Dirichlet series for $\zeta(s)$. Namely, that for $s$ with real part greater than $0$, we have

$$\sum_{n=1}^\infty\frac{1}{n^s}=\zeta(s).$$

So

$$\begin{align*} \sum_{n=1}^\infty\left( \frac{\Gamma(3/2)}{n^{3/2}}-\frac{\Gamma(5/4)}{2n^{5/4}}-\frac{\Gamma(7/6)}{3n^{8/6}}\right ) &=\Gamma(3/2)\sum_{n=1}^\infty\frac{1}{n^{3/2}}-\frac{\Gamma(5/4)}{2}\sum_{n=1}^\infty\frac{1}{n^{5/4}}-\frac{\Gamma(7/6)}{3}\sum_{n=1}^\infty\frac{1}{n^{7/6}}\\ &=\Gamma(3/2)\zeta(3/2)-\frac{\Gamma(5/4)}{2}\zeta(5/4)-\frac{\Gamma(7/6)}{3}\zeta(7/6). \\ \end{align*}$$