4.4: Euler’s Phi or Totient Function
- Page ID
- 60317
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recall the phi function from Definition 4.9.
Lemma 4.15 (Gauss)
For \(n \in \mathbb{N} : n = \sum_{d|n} \varphi (d)\).
- Proof
-
Define \(S(d,n)\) as the set of integers \(m\) between \(1\) and \(n\) such that \(\gcd (m, n) = d\):
\[S(d, n) = \{m \in \mathbb{N} | m \le n \mbox{ and } \gcd (m,n) = d\} \nonumber\]
This is equivalent to
\[S(d, n) = \{m \in \mathbb{N} | m \le n \mbox{ and } \gcd (md, dn) = 1\} \nonumber\]
From the definition of Euler’s phi function, we see that the cardinality \(|S(d,n)|\) of \(S(d, n)\) is given by \(\varphi(\frac{n}{d})\). Thus we obtain:
\[n = \sum_{d|n} |S(d,n)| = \sum_{d|n} \varphi(\frac{d}{n}) \nonumber\]
As \(d\) runs through all divisors of \(n\) in the last sum, so does \(\frac{n}{d}\). Therefore the last sum is equal to \(\sum_{d|n} \varphi(d)\), which proves the lemma.
Theorem 4.16
Let \(\prod_{i=1}^{r} p_{i}^{l_{i}}\) be the prime power factorization of \(n\). Then
\[\varphi (n) = \prod_{i=1}^{r} (1-\frac{1}{p_{i}}) \nonumber\]
- Proof
-
Apply Mobius inversion to Lemma 4.15:
\[\varphi (n) = \sum_{d|n} \mu (d) \frac{n}{d} = n \sum_{d|n} \frac{\mu (d)}{d} \nonumber\]
The functions \(\mu\) and \(d \rightarrow \frac{1}{d}\) are multiplicative. It is easy to see that the product of two multiplicative functions is also multiplicative. Therefore \(\varphi\) is also multiplicative (Proposition 4.3). Thus
\[\varphi (\prod_{i=1}^{r} p_{i}^{l_{i}} = \prod_{i=1}^{r} \varphi (p_{i}^{l_{i}}) \nonumber\]
So it is sufficient to evaluate the function \(\varphi\) on prime powers. Noting that the divisors of the prime power \(p^l\) are \(\{1, p, \cdots p^l\}\), we get from Equation
\[\varphi (p^l) = p^{l} \sum_{j=0}^{l} \frac{\mu (p^j)}{p^j} = p^{l} (1-\frac{1}{p}) \nonumber\]
Substituting this into Equation 4.4 completes the proof.
From this proof we obtain the following corollary.
Corollary 4.17
Euler’s phi function is multiplicative.