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Divisor function $d(n,m)$ counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\equiv0$.

Given $b>0$ what is the correct asymptotic, probabilistic and average case behavior of the function $f(n,m,b)$ that counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\in[-b,b]$?

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There is an evident estimate: $f(n)= \frac{1}{2} \sum_{i=-b}^{i=b} \tau (n+i) + O(b)$ .

P.S. That is for $f(n)=d(n,\sqrt{n})$ in your initial message.

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  • $\begingroup$ can you give a tight estimate? $\endgroup$ – Brout Jun 5 '17 at 10:04
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    $\begingroup$ it is just number of divisors of $n$. $\endgroup$ – Pavel Kozlov Jun 5 '17 at 10:05
  • $\begingroup$ Yeah I know but how does it behave is the question $\endgroup$ – Brout Jun 5 '17 at 10:06
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    $\begingroup$ Tau function has no clear asyptotics like most of multiplicative functions. You can find some upper bounds on it and asymtotics of an average here: terrytao.wordpress.com/2008/09/23/the-divisor-bound. $\endgroup$ – Pavel Kozlov Jun 5 '17 at 11:10

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