I am reading Bernhard Riemann's 1859 paper "On the Number of Primes Less Than a Given Magnitude" in which he derives the Zeta functional equation through a Hankel integral from $+\infty$ around the origin in a clockwise manner back to $+\infty$.
I am familiar with how he derives $$2sin(\pi s)\Gamma(s)\zeta(s)=i\int_{H} \frac{(-x)^{s-1}}{e^x-1}$$ but then he goes to say "If the real part of s is negative, then, instead of being taken in a positive sense around the specified domain, this integral can also be taken in a negative sense around that domain containing all the remaining complex quantities, since the integral taken though values of infinitely large modulus is then infinitely small. However, in the interior of this domain, the integrand has discontinuities only where x becomes equal to a whole multiple of ±2πi, and the integral is thus equal to the sum of the integrals taken in a negative sense around these values."
I know he makes use of the Residue Theorem, but I am more confused on how exactly does he reform the integral in order to have these values contained in the contour?
For $\Re(s) < 0$ $$\int_H \frac{(-x)^{s-1}}{e^x-1}dx= \int_B \frac{(-x)^{s-1}}{e^x-1}dx$$ where $B$ is this contour
to which you can apply the residue theorem