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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 $\tau(x,\frac{1}{2},0)$?

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  • $\begingroup$ Here's the first 100 values of the functions. $\endgroup$ – Mason Dec 8 '18 at 19:01
  • $\begingroup$ I just chose $\tau$ because it's in between $\sigma$ and $\phi$ in the Greek alphabet. It shouldn't be confused with $\tau$ as $\sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon. $\endgroup$ – Mason Dec 8 '18 at 19:10
  • $\begingroup$ Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply. $\endgroup$ – reuns Dec 8 '18 at 20:43
  • $\begingroup$ @reuns. Undoubtedly, I should study the classics. But I think $\tau$ is interesting in that is a type of bridge between important functions. $\endgroup$ – Mason Dec 8 '18 at 20:53
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    $\begingroup$ Since $\sum_{d | n}\phi(d)=n $ then $\sum_{n=1}^\infty \phi(n) n^{-s}= \frac{\zeta(s-1)}{\zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $\sum_{n=1}^\infty \phi(n) n^{-s}$ and $\sum_{n=1}^N \phi(n)$. Can you do the same with your above functions ? $\endgroup$ – reuns Dec 9 '18 at 2:57

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