Summing a multiplicative function

Question

$f(n)$ is a multiplicative function, meaning $f(m\cdot n)=f(m)\cdot f(n)$.

I want to evaluate the sum: $$(1)\qquad\sum_{k=1}^{n}f(m\cdot k)$$ over a fixed $m$. Because $f$ is multiplicative, I can rewrite the sum as: $$(2)\qquad\sum_{k=1}^{n}f(m)\cdot f(k)$$ And because $m$ is fixed, I thought this sum is equal to: $$(3)\qquad f(m)\cdot \sum_{k=1}^{n} f(k)$$ But it's not, as simple paper & pencil check shows. I can easily evaluate the sum $\sum_{k=1}^{n} f(k)$, so I tried to rewrite $(2)$ in terms of it.

For example, suppose $f(n)$ is Euler's totient function, $\varphi(n)$, known as the amount of numbers $<n$ that are comprime to $n$. Suppose $m=4$ and I want to evaluate the sum up to $k=3$, that is $\sum_{k=1}^{3}\varphi(4\cdot k)$. The true sum would be written as $(1)$ & $(2)$ respectively as: $$\varphi(4)+\varphi(8)+\varphi(12) = \varphi(4)\cdot \varphi(1)+\varphi(4)\cdot \varphi(2)+\varphi(4)\cdot \varphi(3) = 10$$

In my wrong approach, the sum would be:

$$\varphi(m)\cdot \sum_{k=1}^{n} \varphi(k)=\varphi(4)\cdot \biggl(\varphi(1)+\varphi(2)+\varphi(3)\biggl)= 2\cdot \space (1+1+2)=8$$ which is different than the original sum. I've noticed this is also true if $f(n)$ is the sum of divisors function too $\sigma_1(n)$, and other multiplicative functions as well.

My question is how can I rewrite $(2)$ in terms of $\sum_{k=1}^{n} f(k)$ or perhaps in another way too, and why is my approach wrong?

Thanks.

Answer 1

The reason you get 8 is a failure of mind: $$2\cdot(1+2+2)=2\cdot 5=10\neq8$$

but $\varphi(2)=1$ , so this sum is completely invalidated.

if not completely multiplicative, as has been talked about, this isn't always going to hold up though. In fact it fails any time $n\geq (p\mid m)$

Answer 2

Let $f$ be multiplicative, but not completely multiplicative, and consider $\sum_{k=2}^2f(2k)$ This is, of course, $f(4)$, and you are asking for a way to modify $(f(2))^2$ to get $f(4)$, or to write $f(4)$ in terms of $f(2)$. This is clearly impossible, since no matter what $f(2)$ is, $f(4)$ could be anything. Multiplicativity is not enough to get an answer to the question you are asking.