Tight bounds for sum of number of divisors (in a range)

Question

If $d(n)$ is the number of divisors of $n$ and we consider: $S(m) = d(1) + d(2) + .... + d(m)$, is it possible to find 2 bounds, $L$ and $U$, based on $log(m)$, $L \le S(m) \le U$ such that $(U - L) \le 1$ at least for $m$ larger than some $m_1$ and smaller than some larger $m_2$. And what results can I use to find by myself similar bounds ?