Let's define $\sigma(n,k) = \sum_{d|n,d\leq k} d$. Notice $\sigma(n,n)=\sigma(n)$, the standard sum of divisor function.
It's a standard exercise to show that $$ \sigma(n) \leq n (\ln n + 1) $$
for some constant $c$.
Is there any known result on the asymptotic when we want to compute the sum of divisors of $n$ up to $k$? Do we expect
$$ \sigma(n,k) \leq k f(n) $$
where $f(n)$ is some small function depending on $n$? Is there any study of this sum? It be great if I can find references, as I need to extend it for something for abelian groups.
It's not hard to see a few partial results. Say if $n$ is a power of a prime, we can let $f(n)=\log_2 n$. If $1,\ldots,k$ are divisors of $n$, then $\sigma(n,k) = {k+1 \choose 2}$. However this implies $n$ is at least the product of all prime numbers between $1$ and $k$, hence $n\sim e^k$. So this seems $f(n) = \ln n$ is sufficient.
So one would conjecture that $\sigma(n,k)\leq k c(\ln n + 1)$ where $c>0$ is some constant.
If one only care about the asymptotic, then $f(n)=o(n^{\epsilon})$ for any $\epsilon>0$. This comes from the number of divisors are at most $n^\epsilon$, and each divisor in the sum can contribute at most $k$.