I have previously asked the question A truncated divisor function sum

where the sum

$$

S_f(x)=sum_{nleq x} min{f(x),d(n)}quad (1)

$$

was of interest, and it was answered satisfactorily.

Here, I am interested in estimating the following quantity

$$

S_a(x,m)=sum_{nleq x} #{d: d|n~mathrm{and}~dleq m}^a

$$

so the divisors are restricted in size, or restricted to the interval $(1,m)$ not in “number” as in (1).

When $a=1,$ this is straightforward (as far as obtaining the main term), since the sum can be evaluated horizontally

$$

S_1(x,m)=sum_{dleq m} lfloor x/d rfloor=left(sum_{dleq m} frac{x}{d}right)+O(m)=n log m + O(m),

$$

and typically I’d be interested in relatively small values of $m$ in terms of $x$.

What about $aneq 1$? In particular, $a=1/2,$ or $a=2,3,$ etc. How can one estimate those sums?