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Let $\operatorname{frac}(n)$ denote the fractional part of a positive integer n. Then, which are the best known bounds for

$$f(n)=\sum_{k=1}^{n} \operatorname{frac}\Big( \frac{n}{k}\Big)$$

?

The only information I could find relates this expression to Dirichlet Divisors Problem. Is there any better bound?

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    $\begingroup$ By trial and error, I have the following crude bounds:$$n\ln n-4n^{1.1}<\sum_{k=1}^n\operatorname{frac}(n/k)<n+n\ln n-3n^{1.1}$$but with no explanation other than trial and error. $\endgroup$
    – Simply Beautiful Art
    Commented Dec 28, 2016 at 0:23
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    $\begingroup$ What is your bound obtained from Dirichlet's divisor problem? $\endgroup$
    – Sungjin Kim
    Commented Dec 28, 2016 at 1:07
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    $\begingroup$ @SimpleArt Your upper bound eventually becomes negative. $\endgroup$
    – Sungjin Kim
    Commented Dec 28, 2016 at 1:09
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    $\begingroup$ @user3141592 Your question is essentially an equivalent one to the Dirichlet's Divisor Problem. So, I wouldn't expect any "better" bound than the one obtained by the DDP. But, still I want to know which bound you obtained so far. $\endgroup$
    – Sungjin Kim
    Commented Dec 28, 2016 at 1:21
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    $\begingroup$ If I understand i707107's comment, you need to combine $ \left\{ y \right\}=y- \lfloor y\rfloor$, with $\sum_{n\leq x}d(n)$, where $d(n)=\sum_{d\mid n}1$ and search in internet about recent progress on Dirichlet Divisor problem (for example the entry of MathWorld). I believe that the more important is that you learn the method of summation if you don't know it (see Murty, Introduction to analytic number theory in YouTube from the offcial channel matsciencechannel, you need to study Lecture 01 Partial summation formula and applications, and at least of lecture 06 from start to minute 24). $\endgroup$
    – user243301
    Commented Dec 28, 2016 at 9:46

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