Does equal Bell series imply Dirichlet convolution?

Apostol states in Ch.$2$ , section $2.17$ of his "Intro to Analytic NT" book;

For any two arithmetical Functions $f$ and $g$ let $h = f*g$. Then for every prime $p$ we have $h_p(x)=f_p(x) \cdot g_p(x)$

Let me explain some of the notation in the more common/known (?) setting, here, arithmetical functions just means $f: \mathbb{N} \to \mathbb{R}$ , here, the symbol $*$ is referring to the Dirichlet convolution, and $f_p(x) $ is referring to the Bell series of $f$ modulo $p$. I know how to prove this, it's quite routine in fact, I want to know if the converse of this holds as well, ie.

If $h_p(x)=f_p(x) \cdot g_p(x)$ for every prime $p$, does this imply $h=f*g$ ?

It is easy to show this is equivalent to asking: If $h(p^n)= f*g \ (p^n)$ for all prime $p$ and $n \in \mathbb{N_{0}}$, does this imply $h=f*g$ ?

I also know that if $f$ and $g$ are multiplicative functions, $f=g \iff f_p(x)=g_p(x)$ for all primes $p$.

So does this mean the answer to my question (in bold) is that the converse holds if (and only if?) $h$ is multiplicative?

I have learnt all this in the past 12 hours only, so I'm not sure if I understand it well at all, but after reading the theorem in Apostol's book, the converse seemed like a natural question to ask, so I did ask myself that question, and I tried to come up with an answer, but I don't think it is satisfactory, mainly because it seems like the thing that the book should have mentioned, as it is very strong(?) and cool and also because I don't think I understand this stuff well right now.

Any answer will be appreciated, Thanks!

Solution 1:

Indeed the identity $h_p=f_pg_p$ is equivalent to $$h(p^n)=(f\ast g)(p^n),$$ for all $n\geq0$. And of course if $f$ and $g$ are multiplicative, this implies $h=f\ast g$.

On the other hand, consider the completely multiplicative functions $g$ and $h$ defined by $g(1)=h(1)=1$ and $$g(p):=0\qquad\text{ and }\qquad h(p):=-1,$$ for every prime number $p$. The function $$f(k):=\begin{cases} (-1)^n&\text{ if $k=p^n$ for some prime $p$ }\\ 2&\text{ otherwise}\end{cases}$$ is clearly not multiplicative, as $f(6)=2$ and $f(2)=f(3)=-1$. Then we have $$(f\ast g)(p^n)=(-1)^n=h(p^n),$$ and so $h_p=f_pg_p$ for every prime number $p$. But for example $$h(6)=1\qquad\text{ and }\qquad (f\ast g)(6)=2,$$ which shows that $h\neq f\ast g$.

Of course this example can easily be adjusted to have $f$ take any values you like outside the prime powers. The simple fact is that the condition that $h_p=f_pg_p$ sets conditions on $f$, $g$ and $h$ only at the prime powers. Even if $g$ and $h$ are both completely multiplicative, if $f$ is not multiplicative then this tells you absolutely nothing about $f$ outside the prime powers.