I'm trying to learn how you can use the Mellin transformation to obtain closed expressions of harmonic sums.
There are demonstrations om MSE how show this technique. eg
Proving $\sum_{n =1,3,5..}^{\infty }\frac{4k \ \sin^2\left(\frac{n}{k}\right)}{n^2}=\pi$
and
Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$
The problem I have is that the identity $\mathcal{M}\left(\sum_{k\geq 1}\lambda_k g(\mu_k x),s\right)=\sum_{k\geq 1}\frac{\lambda_k}{\mu_k}\mathcal{M}\left(g(x),s\right)$. Is only a "real" identity when $k$ ranges over a finite set. But when dealing with an infinite series you only have that it gives an expansion in the neighborhood of 0 (sometimes +$\infty$, depending on the situation).
What do I need to do to prove that harmonic sum identity is true on a certain interval? Trying to proof that you can switch sum and integral doesn't work, because then the equality would be true everywhere for all x. I have looked at the solutions of the above examples, in Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$ the solution is exact for $x\in(0,2\pi)$. Why? How can you formally show this? The argument "because cosine has a period $2\pi$ " is unclear to me.
The Mellin transform technique also requires to calculate the inverse Mellin transform. e.g in previous example you need to calculate the integral $\int_{\frac{5}{2}-i\infty}^{\frac{5}{2}+i\infty} Q(s)/x^s \, ds$. If you do this using the residue theorem with as a contour a rectangle with corners $\frac{5}{2}+iT,\frac{5}{2}-iT,-N-iT, -N+iT$ you will have that if $T\rightarrow \infty$, $N\rightarrow \infty$. only the integral over $(\frac{5}{2}+i\infty,\frac{5}{2}-i \infty)$ remains if $(x\in 2\pi)$. Can this be used to proof exactness on a given interval?
I really find the Mellin transform an interesting and powerful technique, I looked at the survey http://algo.inria.fr/flajolet/Publications/mellin-harm.pdf where they talk about the asymptotics. But now I'm stuck trying to understand how you can have exactness on certain intervals. And how formally to prove this.