Series of gamma function with fixed real part and increasing imaginary part

Question

I'm trying to evaluate the series or to pursue a upper bound, theoretically or numerically: $$ \sum_{k \ge 1} \left| \Gamma(m+2\pi ik/\log q) \right| $$ I know this series is convergent because each term decreases exponentially with k. I tried it with Mathematica, but it says no. And I have done the case with $m=-1,q=2$ here A Fourier series' upper bound involving gamma function. Anyone knows about the universal mean? Thank you for consideration!

Answer 1

Put $y=\frac{2\pi}{\log\left(q\right)} $. Using Stirling approximation we have $$\left|\Gamma\left(m+iky\right)\right|\sim\sqrt{2\pi}e^{-\pi k\left|y\right|/2}\left(k\left|y\right|\right)^{m-1/2} $$ and so $$\sum_{k\geq1}\left|\Gamma\left(m+i\frac{2\pi k}{\log\left(q\right)}\right)\right|\sim\frac{2^{m}\pi^{m}}{\log\left(q\right)^{m-1/2}}\sum_{k\geq1}e^{-\pi^{2}k/\log\left(q\right)}k^{m-1/2} $$ and the series converges if $\log\left(q\right)>0 $. To complete te proof note $$\sum_{k\geq1}e^{-\pi^{2}k/\log\left(q\right)}k^{\alpha}=\textrm{Li}_{-\alpha}\left(e^{-\pi^{2}/\log\left(q\right)}\right)$$ where $\textrm{Li}_{a}\left(z\right)$ is the polylogarithm function.

ADDENDUM. Maybe it's useful to note that this form of the Stirling's approximation of $\left|\Gamma\left(x+iy\right)\right|$ holds if $x_{1}\leq x\leq x_{2} $ for some $x_{1},x_{2} $ and $\left|y\right|\rightarrow\infty. $