Another sum involving totient function or gcd

Question

Motivated by this question, My question pertains to closed form of the sum $$\sum_{d|n}\frac{\phi(d)}{d}$$

There are some formulae and expressions for $\sum_{n}\frac{\phi(n)}{n}$, but what about when we replace $n$ by divisors of $n$ By seeing this question, I think the sum is multiplicative. Any hints? Thanks beforehand.

Answer 1

After a little observation, the sum is similar to evaluating $\sum_{d|n}d\phi(d)$. Since the function $\sum_{d|n}\frac{\phi(d)}{d}$ is (weakly) multiplicative, therefore, it suffices to evaluate the sum for prime powers. We have, $$\sum_{d|p^k}\frac{\phi(d)}{d}=1+\frac{p-1}{p}+\frac{p^2(p-1)}{p^3}+\ldots+\frac{p^k(p-1)}{p^{k+1}}$$ $$=1+k\frac{p-1}{p}$$ Hence the form of the sum for any general $n=\prod_ip_i^{k_i}$ is: $$\sum_{d|n}\frac{\phi(d)}{d}=\prod_{i}\left(1+k_i\frac{p_i-1}{p_i}\right)$$ where $n$ is the product of $p_i$ distinct primes with multiplicity $k_i$

Answer 2

Here is not a closed form, but an alternative representation. Below, $(x,y)={\rm gcd}(x,y)$.

I claim that $$\sum_{k=1}^n (k,n)=n\sum_{d\mid n}\frac{\phi(d)}{d}. $$ To see this, observe that if $d\mid n$ is a divisor of $n$ $$ (k,n)=d \iff \left(\frac{k}{d},\frac{n}{d}\right)=1. $$ In particular, the divisor, $d$, appears exactly $\phi(n/d)$ times in the summation. Thus, $$ \sum_{k=1}^n (k,n)=\sum_{d\mid n}\phi(\frac{n}{d})d. $$ Now, letting $d'=n/d$, the sum can be rewritten as, $$ \sum_{k=1}^n (k,n)=n\sum_{d'\mid n}\frac{\phi(d')}{d'}. $$ Hence, this object has the given operational meaning.

Now, for multiplicative property, take coprime $(m,n)$. Our goal is to show, $$ \left(\sum_{d\mid mn}\frac{\phi(d)}{d}\right) = \left(\sum_{d\mid m}\frac{\phi(d)}{d}\right)\left(\sum_{d\mid n}\frac{\phi(d)}{d}\right). $$ Check that, the left hand side sum has $\phi(mn)=\phi(m)\phi(n)$ terms, so do the right hand side. Furthermore, each divisor $d\mid mn$ can be uniquely decomposed into $d=d_1d_2$ where $d_1\mid m$ and $d_2\mid n$, since $(m,n)=1$. Hence, we deduce the given function is indeed multiplicative.

Remarks: In fact, more is known about this function.

Theorem: For any positive integer $a$, there exists a positive integer $n$, such that, $$ \sum_{d\mid n}\frac{\phi(d)}{d}=a. $$

Try to prove this theorem, it is nice ;)