# Questions tagged [multiplicative-function]

This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.

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### Necessary and sufficient condition to be completely multiplicative

I want to prove that $f*f=f \tau$ iff $f$ is completely multiplicative. The "if" part was relatively easy, using $f(g*h)=(fg)*(fh)$ and plug $g=h=1$ for all $n$. Juxtaposition is ordinary, ...
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### Let $f:\Bbb{N}\to\Bbb{C}$ denotes the indicator function of squares. Express it in terms of Mobious function $\mu$.

Here $f(n)=\begin{cases} 1\ \text{if } n=m^2\text{ for some }m\in\Bbb{N}\\ 0\ \text{if otherwise} \end{cases}$ This is a multiplicative function. At first I define $g:\Bbb{N}\to\Bbb{C}$ be $g(n)$ to ...
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### Fast consecutive prime multiplication

Are there any fast algorithms for multiplying consecutive prime numbers? I found this question about a pattern when multiplying consecutive primes. I was wondering if there is a multiplication ...
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### Why is it important that the $p$-adic absolute value satisfy multiplicativity?

I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like ...
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### $f$ is multiplicative $\implies f^{-1}$ is multiplicative. [closed]

Let $f$ be a multiplicative function i.e. $f(mn)=f(m)f(n)$ for all $m,n$ satisfying $\gcd(m,n)=1$ and $f\not\equiv 0$. Define $f^{-1}$ to be the function $g$ such that $f*g=I$ where $I(n)=1$ if $n=1$ ...
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### Find the values of $f(2)$ for which $f$ cannot be a strictly increasing and completely multiplicative function
I was solving some problems on functional equations especially on multiplicative functions today. I noticed that a strictly increasing completely multiplicative function $f:\mathbb N \to\mathbb N$ ...