I recently ran into the function $f(x)$ defined by the sum $$ f(x) = \sum_{m=1}^{\infty} e^{-x\sqrt{m}}\cos(x\sqrt{m})\, $$ where $x>0$, and I've been unable to make head or tails of it either numerically or analytically.
I've tried a number of things to no avail. The Poisson summation formula fails because the Fourier transform of the summand is more intractable than what we started with. The Euler Maclaurin formula does not apply since the quantity $e^{-x\sqrt{m}}\cos(x\sqrt{m})$ is not differentiable in $m$ at $m = 0$.
Due to its relatively slow convergence, this sum is also hard to evaluate numerically for various values of $x$. All I can say is that numerically it looks like $$ \lim_{x\rightarrow 0_+} f(x) = -1/2\, , $$ but I haven't been able to prove this.
What can be proven about this sum? Does it have a closed form? (I doubt it.) Is there a way to characterize the behavior for small $x$? Is there a more efficient way to evaluate it numerically? Is there some way to approximate it?