Let $\alpha$ be a real number and $|| \cdot ||$ be the distance to the nearest integer. I want to find a non-trivial upper bound for $$ \sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n), $$ which I haven't been able to do yet... Any comments and suggestions or reference are appreciated. Thank you.
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2$\begingroup$ What are you willing to assume about $\alpha$? Because if it is integer (or rational) then nothing better than the trivial $X\log X$ bound is possible. On the other hand if $\alpha$ is a fixed irrational number then one can get $o(X\log X)$ $\endgroup$– Aleksei KulikovCommented Jul 23, 2021 at 17:09
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$\begingroup$ @AlekseiKulikov I was hoping to get an expression that is a minimum of $X \log X$ and something that depends on $\alpha$... $\endgroup$– Johnny T.Commented Jul 23, 2021 at 17:11
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1$\begingroup$ The standard approach is to take a rational approximation $\frac{p}{q}$ to $\alpha$ say one that is guaranteed by Dirichlet's approximation theorem, and then split the sum according to whether $\frac{n p}{q}$ is very close to an integer or not. The right thing to do in this case is probably to split it into consecutive blocks of length $q$, and treat each block seperately as $\frac{X}{n}$ is very different when $n$ is large or small. If I get around to fleshing this argument out I'll add it as an answer. $\endgroup$– RandomCommented Jul 23, 2021 at 22:21
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$\begingroup$ I have found that Lemma 2.2 in Vaughan's The Hardy-Littlewood circle method deals with this sum $\endgroup$– Johnny T.Commented Jul 24, 2021 at 7:45
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$\begingroup$ This is a standard calculation that comes up when dealing with exponential sums in the circle method, so if someone does have a fleshed out argument, it would be good to add it here for the community in case someone wants to reference it later. $\endgroup$– Joshua StuckyCommented Jul 24, 2021 at 14:18
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