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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 the following (Link to the article on page 3):

$$\zeta(s)=\dfrac{\Gamma(1-s)}{2\pi i}\oint\dfrac{t^{s-1}}{e^{-t}-1}dt.$$

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?: enter image description hereenter image description here

Or is this the contour?

enter image description here

For further detail: 1.(h t t p://www.gauge-institute.org/riemann/RiemannZetaFunction.pdf) (pages 21-23 & pages 28-30):

2.(h t t p://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf): ![enter image description here]: pg. 34

3.(h t t p://mathworld.wolfram.com/RiemannZetaFunction.html): p3 g. 25

Thank You!!

Best Regards, J.M

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    $\begingroup$ Edward's book on the Riemann Zeta Function goes over these calculations in detail. In particular, he shows how Riemann derived the first expression you are asking about. $\endgroup$ – Mustafa Said Jul 28 '14 at 19:10
  • $\begingroup$ Thank you, however if you look at a section on pg. 10 (books.google.bs/…), it states that The definition of (-x)^s is (-x)^s= e^(s*ln(-x)), where the definition of ln(-x) conforms to the usual definition of lnz for z not on the real axis as the branch which is real for positive z; thus (-x)^s is not defined on the positive real axis...(Can you please explain this paragraph to me and kindly provide an image of the path described. Also, why must delta be taken to 0 or introduced in the first place? ) $\endgroup$ – VectorCalculus Jul 28 '14 at 23:35
  • $\begingroup$ Thank you, however if you look at a section on pg. 10 (books.google.bs/…), it states that The definition of (-x)^s is (-x)^s= e^(s*ln(-x)), where the definition of ln(-x) conforms to the usual definition of lnz for z not on the real axis as the branch which is real for positive z; thus (-x)^s is not defined on the positive real axis...(Can you please explain this paragraph to me and kindly provide an image of the path described. Also, why must delta be taken to 0 or introduced in the first place? ) @Mustafa Said $\endgroup$ – VectorCalculus Jul 29 '14 at 0:06
  • $\begingroup$ @MustafaSaid (Notification) $\endgroup$ – VectorCalculus Jul 31 '14 at 0:05

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