4.3: Mobius inversion
- Page ID
- 60316
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Lemma 4.12
Define \(\epsilon (n) \equiv \sum_{d|n} \mu (d)\). Then \(\epsilon (1) = 1\) and for all \(n > 1, \epsilon (n) = 0\).
- Proof
-
Lemma 4.8 says that \(\mu\) is multiplicative. Therefore, by Proposition 4.3, \(\epsilon\) is also multiplicative. It follows that \(\epsilon (\pi_{i=1}^{r} p_{i}^{l_{i}})\) can be calculated
by evaluating a product of terms like \(\epsilon (p^{l})\) where \(p\) is prime. For example, when \(p\) is prime, we have
\[\epsilon (p) = \mu (1)+ \mu (p) = 1+(-1) = 0 \nonumber\] and
\[\epsilon (p^2) = \mu (1)+ \mu (p)+ \mu (p^2) = 1-1+0 = 0 \nonumber\]
Thus one sees that \(\epsilon (p^l)\) is zero unless \(l = 0\).
Lemma 4.13
For \(n \in \mathbb{N}\), define
\[S_{n} \equiv \{ (a, b) \in \mathbb{N} | \exists d > 0 \mbox{ such that } d | n \mbox{ and } ab = d \} \nonumber\]
\[T_{n} \equiv \{ (a, b) \in \mathbb{N}^{2} | b | n \mbox{ and } a | \frac{n}{b} \} \nonumber\]
Then \(S_{n} = T_{n}\).
- Proof
-
Suppose \((a, b)\) is in \(S_{n}\). Then \(ab | n\) and so
\[ab = \left. \begin{array} {ab = d}\\ {d|n} \end{array} \right \} \Rightarrow b|n \mbox{ and } a | \frac{n}{d} \nonumber\]
And so \((a, b)\) is in \(T_{n}\). Vice versa, if \((a, b)\) is in \(T_{n}\), then by setting \(d \equiv ab\),
we get
\[\left. \begin{array} {b | n}\\ {a|\frac{n}{b}} \end{array} \right \} \Rightarrow d|n \mbox{ and } ab = d \nonumber\]
And so \((a, b)\) is in \(S_{n}\)
Theorem 4.14: Mobius Inversion
Let \(F : \mathbb{N} \rightarrow \mathbb{C}\) be any number theoretic function. Then the equation
\[F(n) = \sum_{d|n} f(d) \nonumber\]
if and only if \(f : \mathbb{N} \rightarrow \mathbb{C}\) satisfies
\[f(d) = \sum_{a|d} \mu (a) F\left(\frac{d}{a}\right) = \sum_{\{(a,b) | ab = d\}} \mu (a) F(b) \nonumber\]
- Proof
-
\(\Leftarrow\): We show that substituting \(f\) gives \(F\). Define H as
\[H(n) \equiv \sum_{d|n} f(d) = \sum_{d|n} \sum_{a|d} \mu (a) F(\frac{d}{a}) \nonumber\]
Then we need to prove that \(H(n) = F(n)\). This proceeds in three steps. For the first step we write \(ab = d\), so that now
\[H(n) \equiv \sum f(d) = \sum_{d|n} \sum_{ab = d} \mu (a) F(b) \nonumber\]
For the second step we apply Lemma 4.13 to the set over which the summation takes place. This gives:
\[H(n) = \sum_{b|n} \sum_{a|(\frac{n}{b})} \mu (a) F(b) = \sum_{b|n} \left(\sum_{a| \frac{n}{b}} \mu (a)\right) F(b) \nonumber\]
Finally, Lemma 4.12 implies that the term in parentheses equals \(G \left(\frac{n}{b}\right)\). It equals \(0\), except when \(b = n\) when it equals \(1\). The result follows.
Uniqueness: Suppose there are two solutions \(f\) and \(g\). We have:
\[F(n) = \sum_{d|n} f(d) = \sum_{d|n} g(d) \nonumber\]
We show by induction on \(n\) that \(f(n) = g(n)\).
Clearly \(F(1) = f(1) = g(1)\). Now suppose that for \(i \in \{1,\cdots k\}\), we have \(f(i) = g(i)\). Then
\[F(k+1) = (\sum_{d|(k+1), d \le k} f(d)) + f(k+1) = (\sum_{d|(k+1), d \le k} g(d)) + g(k+1) \nonumber\]
So that the desired equality for \(k+1\) follows from the induction hypothesis.