Let $\varphi$ denote the Euler's totient function. Is there any reference in literature for the value of sum $$\sum_{\substack{r\le x\\ d\mid r}}\gcd(\phi(d),r)$$ where $d$ is some fixed positive integer?
Thanks in advance.
Asymptotic for Sum Involving GCD and Euler Totient Function – Number Theory
analytic-number-theorynt.number-theory
Related Solutions
The answer here is
$$\frac{3}{\pi^2}n^2\prod_{p\mid n} \frac{p}{p+1}+O(nd(n)\log\log n+ n\log n),$$
where $d(n)$ is a divisor function and product is over prime values of $p$.
To prove this, note first that
$$\sum_{\substack{a\leq n \\ (a,n)=1}} \varphi(a)=\sum_{a \leq n} \varphi(a)\sum_{d \mid (a,n)} \mu(d)=\sum_{d \mid n} \mu(d)\sum_{bd\leq n} \varphi(bd).$$
Now, using the identity
$$\varphi(a)=a\sum_{d \mid a}\frac{\mu(d)}{d}$$
we get
$$\sum_{bd \leq n} \varphi(bd)=\sum_{b \leq \frac{n}{d}}bd\sum_{c\mid bd} \frac{\mu(c)}{c}.$$
Changing order of summation, we obtain
$$\sum_{bd \leq n}\varphi(bd)=d\sum_{c \leq n}\frac{\mu(c)}{c}\sum_{\substack{b \leq n/d \\ \frac{c}{(c,d)} \mid b}}b.$$
As for any $x$ and $y$ the equality
$$\sum_{\substack{b \leq x \\ y \mid b}} b=\frac{x^2}{2y}+O(x)$$
holds, the inner sum equals
$$\sum_{\substack{b \leq n/d \\ \frac{c}{(c,d)} \mid b}}b=\frac{n^2(c,d)}{2cd^2}+O(\frac{n}{d}).$$
Therefore,
$$\sum_{bd \leq n} \varphi(bd)=\frac{n^2}{d}\sum_{c \leq n} \frac{\mu(c)(c,d)}{2c^2}+O(n\log n).$$
Let us now compute the sum in the RHS of this identity:
$$\sum_{c \leq n} \frac{\mu(c)(c,d)}{c^2}=\sum_{c=1}^{+\infty} \frac{\mu(c)(c,d)}{c^2}-\sum_{c>n} \frac{\mu(c)(c,d)}{c^2}=S_1-S_2$$
(both series are obviously convergent) Now we should compute $S_1$ and estimate $S_2$. Note that the function $\frac{\mu(c)(c,d)}{c^2}$ is multiplicative due to the multiplicativity of $\mu(c), c^2$ and $(c,d)$. Thus,
$$S_1=\prod_p (1-\frac{(c,p)}{p^2})=\prod_p (1-1/p^2)\prod_{p \mid d}\frac{p}{p+1}=\frac{6}{\pi^2}\prod_{p \mid d} \frac{p}{p+1}.$$
To obtain the estimate for the $S_2$, observe that
$$|S_2|\leq \sum_{c>n} \frac{(c,d)}{c^2}\leq \sum_{t \mid d} \sum_{ft>n} \frac{t}{(ft)^2}=O\left(\frac{d(n)}{n}\right).$$
So,
$$\sum_{bd \leq n} \varphi(bd)=\frac{3n^2}{\pi^2d}\prod_{p \mid d} \frac{p}{p+1}+O(nd(n)/d+n\log n).$$
Summing this over all $d$, we finally get
$$\sum_{\substack{a\leq n \\ (a,n)=1}} \varphi(a)=\frac{3n^2}{\pi^2}\sum_{d \mid n} \frac{\mu(d)}{d}\prod_{p \mid d} \frac{p}{p+1}+O(nd(n)\log\log n+n\log n)=\frac{3n^2}{\pi^2}f(n)+O(nd(n)\log\log n+\log n).$$
To get a bit more closed-form expression for $f(n)$ note that $f$ is again multiplicative and for any prime $p$ and positive $\alpha$ we have
$$f(p^\alpha)=\frac{\mu(1)}{1}+\frac{\mu(p)}{p}\frac{p}{p+1}=1-\frac{1}{p+1}=\frac{p}{p+1},$$
therefore
$$f(n)=\prod_{p \mid n} \frac{p}{p+1},$$
as needed.
Best Answer
Let $c:=\gcd(\phi(d),d)$. then setting $r:=dk$ the sum can be rewritten as $$c\sum_{k\leq x/d} \gcd(\tfrac{\phi(d)}c,k) \approx \frac{c^2x}{d\phi(d)}f(\tfrac{\phi(d)}c),$$ where $f(m) := \sum_{k=1}^m \gcd(k,m)$ is the multiplicative function given by its values on prime powers $f(p^s) = p^{s-1}((p-1)s+p)$. See also OEIS A018804.