2.6 Multiplicative Functions

Recall several definitions from classical number theory that we use later. These are useful in studying the density of the intersection of integer circles and describing functions which give this density.

Definition 2.6.1.

A function f:ℤ>0→ℝf:\mathbb{Z}_{>0}\to\mathbb{R} is multiplicative if for all natural numbers a,ba,b with gcd⁡(a,b)=1\gcd(a,b)=1 we have

f⁢(a⁢b)=f⁢(a)⁢f⁢(b).f(ab)=f(a)f(b).
Definition 2.6.2.

Let μ:ℤ>0→{−1,0,1}\mu:\mathbb{Z}_{>0}\to\{-1,0,1\}, the Möbius Function, be defined as follows:

μ⁢(n)={ 1, if n=1;(−1)k, if n=p1⋅p2⁢⋯⁢pk for pi≠pj while i≠j; 0, if there exists an integer d>1 such that d2|n.\mu(n)=\left\{\begin{array}[]{l}\ \ 1,\hbox{ if $n=1$};\\ (-1)^{k},\hbox{ if $n=p_{1}\cdot p_{2}\cdots p_{k}$ for $p_{i}\neq p_{j}$ % while $i\neq j$};\\ \ \ 0,\hbox{ if there exists an integer $d>1$ such that $d^{2}|n$.}\end{array}\right.
Definition 2.6.3.

Let τ:ℤ>0→ℤ>0\tau:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0} be the number of divisors function defined as:

τ⁢(n)=∑d|n1.\tau(n)=\sum_{d|n}1.
Remark 2.6.4.

The Möbius function μ\mu and number of divisors function τ\tau are both multiplicative.

Definition 2.6.5.

Let us consider a family F={fi|i∈I}F=\{f_{i}|i\in I\} of statements parametrised by the set of indices II. Then the predicate is a binary function of II defined as follows:

[fi]={1⁢ , if fi is true;0⁢ , if fi is false.[f_{i}]=\left\{\begin{array}[]{l}1\text{ ,\quad if $f_{i}$ is true;}\\ 0\text{ ,\quad if $f_{i}$ is false.}\end{array}\right.