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Hi Math Stack Exchange,

I'm working on a problem that involves the difference between a sum series of floor functions. I have tried taking the more standard number theory approach by looking at remainder classes and modular arithmetic but haven't had real success so I'm hoping to take an analytical approach and was looking for help.

Let,

$$f(L) = \sum_{k=2}^{\frac{L}{2}}{\lfloor{\frac{L}{k}}\rfloor}$$

and

$$\Delta(c,L) = f(L+c) - f(L) = \sum_{k=2}^{\frac{L}{2}}{\lfloor{\frac{L+c}{k}\rfloor - \lfloor\frac{L}{k}}\rfloor}$$

For the problem we can assume c is an integer and is very very small in comparison to L. So given the above equations I have a couple questions and any help on any of them would be great!

1) Does there already exist a quick identity for f(L) or $\Delta(c,L)$?

2) If there doesn't exist a nice identity, is there an analytical way to estimate f(L) or $\Delta(c,L)$ or approximate them?

Thank you guys for taking the time to look over this!

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Your $f(L)$ is very closely related to the classically studied $$ D(X) = \sum_{n \leq X} \left\lfloor \frac{X}{n} \right\rfloor = \sum_{n \leq X} d(n) = X \log X + c X + O(X^{1/3})$$ for a known constant $c$. This is sometimes called 'Dirichlet Divisor problem', and more information can be found on the Divisor summatory problem page on wikipedia. I note that the $X^{1/3}$ is actually conjecturally $X^{1/4}$, maybe with some log factors.

Then your $f(L) = D(L) - L$. You can't really hope to understand $f(L)$ better than the masses of men who have tried in quiet desperation to improve estimates for $D(L)$. The key phrase to get you started learning about these techniques is to look up "the hyperbola method" for the divisor problem.

This asymptotic will also give you the leading order asymptotic for $\Delta(c, L)$.

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  • $\begingroup$ Thanks! This was really helpful. Do you know if there is a generalized form of this for the case where its $$\sum_{jk\leq X} \lfloor \frac{X}{jk} \rfloor$$? $\endgroup$ – user122523 Jan 1 '18 at 16:49

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