Suppose I knew the behaviour of a given sum in every other interval, for example: $$ \sum_{\substack{0\leq a \leq x\\ a\equiv 1 (k)}} \sum_{x/(a+k/2)< b \leq x/a} f(b) \sim g(x), $$ for any $x>1$, and fixed integer $k$, and $f$ an arithmetic function with $|f(a)|\leq 1$, then is it possible to say that the complete sum $$ \sum_{0\leq n \leq x} f(n) \sim P(g(x)) $$ where $P$ is some explicit function of $g(x)$ also? It seems like something this or very close to it should be true, but I am not certain what the right statement should be.
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$\begingroup$ This sounds like a question for Fourier methods (of which I know nothing, so use some skepticism on this comment). If f behaved nicely on your test intervals and [something mumble something] with the Fourier transform, then you might be able to get good bounds for your estimate. If f oscillated wildly on the intervals outside of your test range, hopefully the transform of f could give you a picture of what to expect. Gerhard "Hopes An Expert Chimes In" Paseman, 2017.04.02. $\endgroup$– Gerhard PasemanCommented Apr 2, 2017 at 20:57
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$\begingroup$ This seems connected somehow to the Nyman–Beurling criterion for the Riemann hypothesis: there the fractional part function (triangular waves) are used instead of square waves here, but the joint general idea of assembling a linear combination of the waves into a constant function is intriguing. $\endgroup$– Greg MartinCommented Apr 2, 2017 at 21:37
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