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 $n\bmod q\in[-b,b]$?