This afternoon I was reading a lecture (see below) about Fourier transform, and Poisson summation. I've asked myself this
Question. Let $f(x)$ a Schwartz function, and $\mu(n)$ the Möbius function. Is it possible to study the asymptotic behaviour of $$\sum_{k=1}^N\left(\mu(k)\sum_{n=-k}^k f(n)\right)$$ as $N\to\infty$? Thanks in advance.
The vagueness in previous words Is it possible to study, is due that I don't know if it is feasible, to state a proposition about the asymptotic behaviour as an asymptotic equivalence or with big oh notation, for the set of all Schwartz functions (I don't know if it is interesting or was in the literature, but seems nice).
My only attempt was by means of partial summation (Abel's Lemma) to get $$\sum_{k=1}^N\left(\sum_{n=-k}^k f(n)\right)\cdot \mu(k)=\left(\sum_{n=-N}^N f(n)\right)M(N)-\sum_{k=1}^{N-1}M(k)(f(-k-1)+f(k+1)),$$ where $M(N)=\sum_{n=1}^N\mu(n)$ is the Mertens function.
The order or magnitude of the Mertens function is an unsolved problem, but are known unconditional statements. If my question has mathematical meaning I need to implement the definition to be a Schwartz function (the condition that I refer is in DEFINITON 3 of [1], but if you work with a different condition/definittion of Schwartz function I will accept it). Also I don't know if it's possible to exploit Poisson with this purpose.
Does make mathematical meaning my question? I say to study the asymptotic behaviour of our arithmetic function for all Schwartz function, by means of a result/proposition.
References:
[1] Candelori, Fourier transforms and Poisson summation (2011). Is a lecture currently available from this page of McGill University.