The following corollary demonstrates that in order to calculate the value $\beta(n)$ of a multiplicative function $\beta:\mathbb N\to\mathbb C$ it is sufficient to know the values of $\beta(p^e)$ on all maximal powers of prime numbers $p^e$ with $p^e\mid n.$# Corollary: Simple Conclusions For Multiplicative Functions

(related to Definition: Multiplicative Functions)

For all multiplicative functions $\beta:\mathbb N\to\mathbb C$ we have $\beta(1)=1$ and $$\beta(n)=\prod_{i=1}^\infty{\beta(p_i^{e_i})}$$ for the factorization of $n=\prod_{i=1}^\infty p_i^{e_i}.$

Proofs: 1

Examples: 1

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  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927