Let $$B(s)=\sum_{n\le N}b(n)n^{-s}$$ with $b(n)\ll n^{\varepsilon}.$ I want to study $$\frac{1}{2\pi i}\int_C\frac{\zeta'}{\zeta}(s)\zeta'(s)\zeta'(1-s)B(s)B(1-s)\;ds,$$ with $C$ being the contour with vertices $1+1/\log T+i, 1+1/\log T+iT, -1/\log T+iT, -1/\log T+i.$ Something of this form was studied in the 90s (see $S_2$ here) but they formulate the polynomial $B$ slightly differently, and I want to find an asymptotic which can later be specialised to the 'correct' answer simply by substituting in the choices of coefficients $b(k)$.
The reason I ask this is if, for example, I wanted $B(s)$ to resemble something like $\zeta'(s)^2$ I would choose $b(n)=(\log*\log)(n)$ (and then $B(s)$ would just be the truncated Dirichlet series of $\zeta'(s)^2$), but it doesn't seem that one can make such a choice in the paper? Any insights?
