I am trying to do contour integration on $\int_c \zeta(s)\zeta(2s) \frac{x^s}{s} ds$ and $\int_c \frac{x^s}{s^ks}ds$ where c is the line segment joining c-iT c+iT
I understand the basic theory behind it, how to find residues and Cauchy's theorem. I am just having trouble with bounding these integrals when using Cauchy's theorem.
What I mean to say here is let's say I am doing contour integration for x>1 on a square say for $\int_c \frac{x^s}{s}ds$ I can bound the horizontal integral by $\frac{1}{2\pi T}\frac{y^c}{log(y)}$ can I bound the horizontal integral for $\int_c \zeta(s)\zeta(2s) \frac{x^s}{s^ks} ds$ by $\frac{1}{2\pi T^kT}\frac{y^c}{log(y)}$?