Let's consider: $$\tau(x,a,b)=\sum_{1 \le d \le x \\ (d,x)=d^a \\} d^b$$
Where $(q,r)$ denotes the gcd of $q$ and $r$.
I think this could be interesting thing to look at because it's somehow a type of bridge between the sum of the divisor function $\sigma_k(x)=\tau(x,1,k)$ and Euler's totient function $\phi(x)=\tau(x,0,0)$.
Now, the average order of these functions is fairly well understood.
For example,
$$\sum_{n=1}^x \tau(n,1,1) \approx \frac{\pi^2}{12}x^2$$
$$\sum_{n=1}^x \tau(n,0,1) \approx \frac{1}{\pi^2}x^3$$
$$\sum_{n=1}^x \tau(n,1,0) \approx x\log(x)+(2\gamma+1)x$$
$$\sum_{n=1}^x \tau(n,0,0) \approx \frac{3}{\pi^2}x^2$$
And these can be argued using the standard techniques which are Abel Summation formula and Dirichlet convolutions. Where $\gamma$ is the Euler Macheroni constant.
Is it possible to achieve similar results for non integer values $a$?
For example, what is the average order of $\tau(x,\frac{1}{2},1)$? What is the average order $\tau(x,\frac{1}{2},0)$?