I need to find the following limiting value: $$lim_{T\to\infty}\int_{2-iT}^{2+iT} \frac{a^s}{s}ds$$ where $s \in \mathbb{C}$ and $a\in \mathbb{R}$
This came up in the context of trying to understand the distribution of primes with respect to the zeros of the Riemann Zeta Function. You might find some printing mistakes, but you may refer to 'A History of PNT' by LJ Goldstein for the same.
Thank you in advance
Let $C_T$ be the rectangle $[-T,2]+i[-T,T]$. Then for $a > 1$ $$\lim_{T\to \infty} \frac1{2i\pi}\int_{\partial \ C_T} \frac{a^s}{s}ds=lim_{T\to\infty}\frac1{2i\pi}\int_{2-iT}^{2+iT} \frac{a^s}{s}ds$$ On the other hand $$\frac1{2i\pi}\int_{\partial \ C_T} \frac{a^s}{s}ds=Res(\frac{a^s}{s},0)=1$$ For $a \in (0,1)$ it is the same with the rectangle $R_T=[2,T]+i[-T,T]$, it contains no pole thus $\frac1{2i\pi}\int_{\partial \ R_T} \frac{a^s}{s}ds=0$.