How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results."
The contour integral is stated as: $$\zeta(s)=\dfrac{\Gamma(1-s)}{2\pi i}\oint\dfrac{t^{s-1}}{e^{-t}-1}dt.$$
(Page-3of this PDF)
where the contour of integration encloses the negative $t$-axis, looping from $t = −∞ − i0$ to $t = −∞ + i0$ enclosing the point $t=0$. The equivalence of $(2.11)$ and $(2.1)$ is well-described in texts (e.g. [35]) and is obtained by reducing three components of the contour in $(2.11)$. A different analysis is possible however (e.g. [12], Section 1.6) - open and translate the contour in $(2.11)$ such that it lies vertically to the right of the origin in the complex $t$-plane2 with $σ < 0$ and evaluate the residues of each of the poles of the integrand lying at $t = ± 2nπi$ to find:
$$\zeta(s)=\dfrac i{2\pi}\Gamma(1-s)(2\pi i)^s\big(\exp(i\pi s/2)-\exp(-i\pi s/2)\big)\sum_{k=1}^\infty k^{s-1}.$$
Can you please explain the equivalence of the first expression and the Riemann Zeta function, as [35] does not explicitly state the correlation. In the first expression, what does t represent (is it a complex or real variable?). On page 22 of the The Riemann Hypothesis Origin, it states: to define the multi-valued function (-λ)^(z-1)=e^((z-1)*ln(-λ)), we choose the branch of ln(−λ) that is real for λ < 0. (In our case −λ=t and z=s) In addition, how would one evaluate this complex (assuming t is complex, however knowing s can be complex) contour integral? Would one apply the method of branch cuts or the Residue Theorem, and if so; can you demonstrate it, as I am relatively new to the concept. Most importantly, can you also identify the contour being integrated over.
Is this the contour being integrated over?
Or is this the contour?
For further details
http://www.gauge-institute.org/riemann/RiemannZetaFunction.pdf
(Pages21-23& Pages28-30)http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf
(Page34)http://mathworld.wolfram.com/RiemannZetaFunction.html
(p3Page25)