I'm looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously):
$ \sigma_0(n,m) = \sum_{d|n,d<m} 1 $.
Or more generally:
$ \sigma_x(n,m) = \sum_{d|n,d<m} d^x $.
Or any important property of either (like parity).