How do I evaluate the sum $$\sum_{0 <\gamma \leq T} x^{i\gamma}, $$ where $\gamma$ is the imaginary part of a non-trivial zero of Riemann's zeta function and $x, T>0$?
My attempt is \begin{equation*} \begin{split} \sum_{0 <\gamma \leq T} x^{i\gamma} &=\sum_{0 <\gamma \leq T} (\cos(\gamma \log x)+i\sin(\gamma \log x)) = \int_{0}^{T} \cos(\gamma \log x) \mathrm{d}N(t)+i\int_{0}^{T} \sin(\gamma \log x) \mathrm{d}N(t), \end{split} \end{equation*} where $N(t)=O(t\log t)$ is the number of non-trivial zeros between $0$ and $t$. By change of variables, $$\int_{0}^{T} \cos(t \log x) \mathrm{d}N(t)+i\int_{0}^{T} \sin(t\log x) \mathrm{d}N(t)=\int_{0}^{T} \cos(t \log x)\log t \mathrm{d}t+i\int_{0}^{T} \sin(t \log x) \log t \mathrm{d}t.$$ But the above integrals do not have a plain solution.
if RH is true one has for fixed $x>1$ and $T>T_x$ that $$\sum_{0 <\gamma \leq T} x^{i\gamma}=-\frac{T}{2\pi}\frac{\Lambda(x)}{\sqrt x}+\frac{x^{iT}\log (T/2x)}{2\pi i \log x}+O(\frac{\log T}{\log \log T})$$ while there is a formula more complicated that is universal in $x, T$ but the main term is still $-\frac{T}{2\pi}\frac{\Lambda(x)}{\sqrt x}$, where as usual $\Lambda(x)=\log p, x=p^n, n \ge 1$ and $0$ otherwise
See Fujii, On A Theorem of Landau II
Unconditionally one can estimate $$\sum_{0 <\gamma \leq T} x^{\beta+i\gamma}=-\frac{T}{2\pi}\Lambda(x)+O(\log T)$$ again for $x>1$ fixed and $T>T_x$ and a more complicated expression that is universal in $(x,T)$ but now we cannot say much more about $$\sum_{0 <\gamma \leq T} x^{i\gamma}$$ since we may have lots of non trivial zeroes with $\beta \to 0$ for which the two sums (with $\beta+i\gamma$ and $i\gamma$ only) are similar
See Fujii, On A Theorem of Landau I and references there