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Is it possible to pick up the value $\log^2 x<y<x$ to get this expressions simultaneously $$\sum_{d\leq y}\left \{ \frac{x}{d} \right \}\leqslant \frac{y}{\log^{3} y}?$$

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    $\begingroup$ please specify the quantifiers for $x, y$, the words "pick up" and "simultaneously" seem contradictory for me $\endgroup$
    – Fedor Petrov
    Commented Oct 20, 2022 at 9:01
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    $\begingroup$ The estimate in your question implies that the numbers $\{\frac{x}{d}\}$ would be biased to be very close to zero, that is $$ \left\{\frac{x}{d}\right\} \leq \frac{1}{(\log y)^3} $$ for many $d$, but there is (as far as I know) absolutely no reason to believe this to be the case, and least when $y$ is smaller than $x^{1/2-\varepsilon}$, say. $\endgroup$
    – Joshua Stucky
    Commented Oct 20, 2022 at 20:35

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