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Example: Examples of Multiplicative Functions

(related to Definition: Multiplicative Functions)

• The prime-counting function $\pi\mathbb N\to\mathbb N$ and the von Mangoldt function $\Lambda:\mathbb N\to\mathbb N$ are obviously not multiplicative. For instance, $\pi(1)=0\neq 1$ and $\Lambda(1)=0\neq 1,$ which would otherwise be the case because of the above corollary.
• The number of divisors function $\tau\mathbb N\to\mathbb N$ is multiplicative. In particular, we have $\tau(1)=1$ and, for instance $\tau(24)=\tau(2^3)\cdot\tau(3^1)=4\cdot 2=8.$
• The Moebius function $\mu:\mathbb N\to\mathbb N$ is multiplicative, since if $n,m$ are co-prime and square-free, then their product is square-free. Therefore $\mu(nm)=\mu(n)\mu(m).$ Moreover, we have $\mu(1)=1$ by definition.
• The Euler function $\phi:\mathbb N\to\mathbb N$ is multiplicative. The argument for this proposition is the following:
  • For any number $n\ge 1$, there are $\phi(n)$ numbers co-prime to $n$ among the numbers $1,\ldots,n$, also $\phi(n)$ numbers co-prime to $n$ among the numbers $n+1,\ldots,2n$, also $\phi(n)$ numbers co-prime to $n$ among the numbers $2n+1,\ldots,3n,$ also $\phi(n)$ numbers co-prime to $n$ among the numbers $3n+1,\ldots,4n,$ etc.
  • If we have $m\ge 1$ and want count the numbers co-prime to $nm$ among the numbers $1,\ldots,nm$ then we shall start with counting the $m\cdot \phi(n)$ numbers co-prime to $n$ among the numbers $1,\ldots,nm$ and then remove those numbers, which are multiples of $m$.
  • This is equivalent to saying that $\phi(n)\cdot \phi(m)$ is the count of the numbers $1,\ldots,nm$ which are co-prime to both, $n$ and $m$. This, by definitions, equals $\phi(nm).$

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927