Proof
(related to Corollary: Simple Conclusions For Multiplicative Functions)
- Let $\beta:\mathbb N\to\mathbb C$ be multiplicative.
- Then there exists $m\in\mathbb N$ with $\beta(m)\neq 0.$
- Because the multiplication of complex numbers is cancellative, it follows from \(\beta(m)\beta(1)=\beta(m)\cdot 1\) that \(\beta(1)=1.\)
- Moreover, if $n=\prod_{i=1}^\infty p_i^{e_i}$ is the factorization of $n,$ then the greatest common divisor $\gcd(p_i^{e_i},p_j^{e_j})=1$ for any two prime numbers $p_i\neq p_j.$
- Therefore, the factors $p_i^{e_i}$ and $p_j^{e_j}$ are co-prime.
- From the multiplicativity of $\beta$ it follows $\beta(n)=\prod_{i=1}^\infty{\beta(p_i^{e_i})}.$
∎
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
References
Bibliography
- Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
