In this paper https://www.biodiversitylibrary.org/item/93357#page/65/mode/1up of Landau he wants to evaluate $$\int _{2-iT}^{2+iT}\frac {x^s\log (s-1)}{s^2}\hspace {10mm}(1)$$ (page 755). He sets up a keyhole-type contour path, something like $$\int _{\text {path from $2+iT$ to $1/2$}}+\int _{L_1}+\int _{L_2}+\int _{\text {path from $1/2$ to $2-iT$}}\hspace {10mm}(2)$$ where $L_1,L_2$ are paths between 1/2 and 1, essentially on the real line. He then applies Cauchy's Residue Theorem - he says (1) and (2) are equal.
I feel I understand the general idea, and understand that we choose a different argument for $\log (s-1)$ when we take the path the second time, and that it's this $2\pi i$ increment that gives us a main term. (I also understand that each individual integral exists because $\log $ growth means we can take the integrals up to the singular point.) Still, when I'm pushed to write down concretely and precisely what's happening then I find I can't do it.
Specifically:
1.Cauchy's Integral Theorem, as far as I know, is a statement about functions on a subset of $\mathbb C$. In that case, what exactly is the subset we take above? Is it $\mathbb C-\mathbb R^{\leq 0}$ for example? It could well be that I'm oversimplifying and that it's not just a subset.
What precisely are the paths $L_1$ and $L_2$? Are they literally on the real axis or are they just above and below, with an $\epsilon $ argument that's not written down explicitly?
Does the path self-intersect along the real axis, and if so how do we apply Cauchy's Theorem?
I'd like it if someone could tell me precisely where/how Cauchy's Integral Theorem is being applied, perhaps having to be very careful about the exact statement.
Sorry for being so pedantic it just seems to be something I need to get clear! Thanks:)
When $\log$ is meant to take the principal value, the function $s^{-2}x^s\log(s-1)$ is holomorphic and one-valued inside and on the contour described as follows.
Consequently, Cauchy's theorem applies, and notice that the integral over the circle around $s=1$ vanishes when the radius of the circle tends to zero, so you will obtain the desired result.