Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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70 views

How can I prove this functional equation?

Question:- If $$F(x)=f(x)+\frac12f(x^2)+\frac13f(x^3)+\frac14f(x^4)+\ldots\,,$$ then prove that $$f(x)=F(x)-\frac12F(x^2)-\frac13F(x^3)-\frac15F(x^5)+\frac16F(x^6)-\frac17F(x^7)+\frac1{10}F(x^{10})-\...
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2answers
56 views

Finding $f$ with Möbius inversion formula

How to find an arithmetic function $f$, when the summatory function $F$ of $f$ is given by $F(n)=\begin{cases} 1 & \mathrm{if}\ n \mathrm{\ is \ a \ square \ number} \\ 0 & \mathrm{...
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0answers
41 views

Mobius inversion on the partition lattice

For some $n \in \mathbb N$, let $(\Pi_n, \le)$ be the poset of partitions of the set $\{1, 2, \dots, n\}$, where two partitions $\pi, \rho \in \Pi_n$ have the relation $\pi \le \rho$ if $\pi$ is a ...
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44 views

Correspondence between $k$-ary Lyndon words and $(k-1)$-ary Lyndon words without repetitions

At Counting Lyndon words with no adjacent character repeats, it turned out that for $n\ge3$ the number of $k$-ary Lyndon words of length $n$ without adjacent identical letters is the number of $(k-1)$-...
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1answer
35 views

Proof of an identity concerning the prime $\zeta$ function

I have to prove the following identity: let $P(s)=\sum_p\frac{1}{p^s}$, for $Re(s)>1$, then \begin{equation} P(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log(\zeta(ns)). \end{equation} I proved that \...
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1answer
63 views

Prove that the lower bound for the partial sums of the Möbius inverse of the Harmonic numbers minus $n$ is greater than $1$ minus a Harmonic number

Let $\varphi^{-1}(n)$ be the Dirichlet inverse of the Euler totient function: $$\varphi^{-1}(n) = \sum\limits_{d\mid n} d \cdot \mu(d) \tag{1}$$ and let the matrix $T$ be: $$T(n,k)=\varphi^{-1}(\...
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30 views

Do we know the Mobius inverse of e^x?

I'm refereing to the The Generalized Möbius Inversion: Suppose F(x) and G(x) are complex-valued functions defined on the interval [1,∞) such that $$ G(x)=\sum _{1\leq n\leq x}F\left({\frac {x}{n}}\...
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1answer
35 views

Value of Möbius series involving $\log(n+1)$

Let $\mu$ be the Möbius function. It is known that $$\sum_{n \geq 1} \frac{\mu(n)}{n} \log n = -1$$ I'm looking for a closed-form expression of the similar series $$\sum_{n \geq 1} \frac{\mu(n)}{n} ...
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39 views

Question on proof of relationships between $f(s)=\frac{s}{s+1}$ and the analytic Harmonic number function $H(s)$

This question assumes the following definitions where $s\in\mathbb{C}$. (1) $\quad f(s)=\frac{s}{s+1}$ (2) $\quad H(s)=\psi(s+1)+\gamma\qquad\text{(analytic harmonic number function)}$ Question: ...
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Question on the coefficient of the Dirichlet series related to $\frac{\zeta(s+2)}{\zeta(s)}$

This question is about the evaluation of $a(n)$ defined in (1) below which is related to the Riemann zeta function $\zeta(s)$ as illustrated in (2) below. (1) $\quad a(n)=\sum\limits_{d|n}\frac{\mu\...
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12 views

Power of a point on $\mathbf{P}^1$

Is it possible to define the power of the point on $\mathbb{E}^2$ to be invariant under projective transformations? Although it is a mostly metrical concept, perhaps it is possible to extend this ...
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1answer
51 views

Dirichlet Self-Convolution Inversion

I am interested in finding out a method to invert Dirichlet selfconvolution. In math expressions it means: Find out $a$ once $b=a*a$ is known So a kind of squareroot of the Dirichlet product. I ...
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2answers
149 views

Möbius function and Posets

As usual, $\mathbb{C}$ denote the field of complex numbers. Let $\mu \in I_{\mathbb{C}}(P)$ (the $\mathbb{C}$ incidence-algebra of $P(X, \leq)$ a poset). I am asked to show the following are ...
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1answer
34 views

$\Phi_{2^n}(x) = x^{2^{n-1}}+1$ by Mobius Inversion

I want to prove that $\Phi_{2^n}(x) = x^{2^{n-1}}+1$. This is not hard to do by using recursion. However, I want to know if there's a simpler way to do this using the Mobius Inversion formula, which ...
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1answer
71 views

Squarefree integers and floor function, Möbius function

On page 40, exercise 44 of Introduction to Analytic and Probabilistic Number Theory by Tenenbaum: Show that any integer $n\ge1$ can be uniquely decomposed as $n = qm^2$ , where $q$ is squarefree. ...
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37 views

Number of subsets of size $k$ with a given GCD

Given a set $S = \left\{1, 2, \ldots,N \right\}$ of positive integers, we want to count the number of subsets of size $k$ with the GCD of all elements (henceforth referred to as the GCD of the set) ...
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1answer
76 views

Need help with Möbius function

Suppose I need to find total subset of numbers of length $K$ in range $1$ to $N$ such that their $\gcd$ is $g$. How can I utilize Möbius function for that. So approach is we can choose $K$ numbers ...
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51 views

Arbelos - Pappus chain radius of tangent circles

I would like to determine radius' of pappus chain circles which are tangent to Arbelos. Below I will put some photos from one book that showed how to determine it. The only thing that i don't ...
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1answer
33 views

which domain does this Möbius map to $\Re(w) > 0$?

Given the following Möbius: $$ w = T(z) = \frac{1+z}{1-z} $$ How could I find the domain of $Z$ which $T$ maps to $\{\Re(w)>0\} $? I tried to inverse $T$ and got: $$ z = T^{-1}=\frac{w-1}{w+1} $$ ...
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1answer
49 views

Continuation of the Riemann Prime Counting function

Let $f(x)=\sum_{n=1}^{\lfloor \ln(x)\rfloor} \frac {\pi(x^{\frac 1n})}n$ ($\pi(x)$ is the Riemann prime counting function, here are some more informations). It is thanks to Riemann well known that $\...
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1answer
84 views

Inversive distance between concentric circles

I found an explication about Inversive distance in Coxeter book. But, i don't really understand some things in its explication. I can't imagine the meaning of: ''This inquiry almost forces us to ...
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30 views

Construction of inverse point in sphere

i would like to know if my construction of inverse point of A in sphere, A' is right. First of all i constructed a line through pole $S$ and $A$, then i drew a line through $N$ perpendicular to $AS$. ...
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1answer
84 views

Mobius inversion problem [closed]

Prove by Mobius inversion formula if $\frac{n}{\phi(n)}=\sum_{d\mid n} f(d)$ then $f(d)=\frac{\mu^2(d)}{\phi(d)}.$
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1answer
113 views

Number of Regions for a Central Hyperplane Arrangement

This question has likely been answered in full detail before, so any references would be greatly beneficial. The question I have is as follows: Suppose we have $m$ central hyperplanes in $\mathbb{R}^...
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1answer
197 views

Prove that this linear programming problem is bounded by $O(k^{1/2})$

The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems: $$\begin{array}{ll} \text{minimize} &...
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2answers
63 views

How to write $\zeta(s)^3$ as a function of the generalized Mobius function?

Popovici (1963) (see link), created a way to extend the Mobius function, $\mu(n)$, to the complex plane. The Mobius $\mu(n)$ function is such that: $\frac{1}{\zeta{(s)}}=\sum_{n=1}^{\infty}\frac{\mu(...
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1answer
44 views

Simplifying expression Mobius Function

Can anybody help me simplify this expression using Mobius Inversion Formula or any other result in order to calculate F(3500) in a simple way?? $$F(n)=\sum_{d\mid n} \mu(d)d$$
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1answer
39 views

Two sequence relations are equivalent.

For two sequences of complex numbers $\{a_0, a_1, ..., a_n, ...\}$ and $\{b_0, b_1, ..., b_n, ...\}.$ Show that the following relations are equivalent: \begin{equation} a_n = \sum_{k=0}^n b_k \iff b_n ...
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1answer
79 views

Number of words with length $n$ and symbols $\{1,2,3\}$ which are not equal to any of their cyclic shifts, with Möbius Inversion.

Let $t_{n}$ be the number of words with length $n$ and symbols $\{1,2,3\}$ which are not equal to any of their cyclic shifts. For example, the word $1232$ has cyclic shifts $2123, 3212, 2321$, none of ...
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1answer
69 views

Easy way to prove property of prime indicator

Let $\mu$ be the Möbius function, and let $\nu(n)$ be the number of distinct prime factors of $n$. Then we can define $p = \mu * \nu$, i.e. $$ p(n) = \sum_{d \mid m} \mu(d) \nu(n/d). $$ An exercise in ...
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1answer
310 views

Proof of Mobius inversion formula - the other direction

The Mobius inversion formula states that if we define $f$ as $$f(m)=\sum_{d|m}g(d)$$ then $$g(m)=\sum_{d|m}f\left(\frac md\right)\mu(d)$$ We know that the other direction is also true: if we define $...
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1answer
92 views

How to show that $\sum_{d\mid n} \gcd(d,k) \mu(n/d)=0$?

Fix $k\in \Bbb{N}$ such that $k > 1$. Define $f_k(n)=\gcd(n,k)$ for $\forall\ n \geq 1$. Let $\mu$ denote the Möbius function. Notice that $$\sum_{d\mid n}f_k(d)\mu\left(\frac{n}{d}\right)=0$$ for ...
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1answer
69 views

Mobius transformation producing a curved triangle with 3 intersecting circles

Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume ...
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$f(n) = \Sigma_{d|n} \mu(n/d)F(d)$

The question says: If $F(n) = \Sigma_{d|n} f(d)$ for every positive integer $n$, prove that $f(n) = \Sigma_{d|n} \mu(n/d)F(d)$. What I know so far is that divisors of $n$ can be paired together. ...
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1answer
197 views

Proving relationship between Euler's totient function and Mobius function

I'm trying to prove that $$\phi(n)=\sum_{d|n}\mu(d)\phi(d)$$ My attempt is the following: Mobius inversion formula tells us that, since $$n=\sum_{d|n}\phi(d)$$ then $$\phi(n)=\sum_{d|n}\mu(d)\...
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1answer
65 views

A functional ecuation using divizors

It’s a functional equation. We have a function f defined on pozitive integers (greater than 0) with values on real numbers. Also, for any pozitive (and non zero) integer n. It asks to find function f. ...
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1answer
289 views

Reasoning about factorials: is this equation correct?

I was playing around with the möbius inversion formula and factorials and I came up with an equation that I found interesting: $$\prod_{i\ge2}\left\lfloor\frac{x}{i}\right\rfloor! = \prod_{i \ge 2}\...
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Möbius inversion across all natural numbers (no divisors)

EDITED TO ACCOMMODATE COMMENTS: I'm trying (self-taught) to understand more about Möbius inversion. Take two arithmetic functions $f$ and $g$ defined by $$g(n)=\sum_{d|n}f(d)$$ (Presumably $d$ ...
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How to invert an arithmetic function where Möbius inversion may not apply?

I'm playing around with problems in order to gain a very basic insight into number theory, and I am looking at the process of inversion. Take a convergent arithmetic function of the general form $$f(...
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1answer
233 views

Identity involving the Möbius function and the first derivative of the Riemann zeta function

Working on the derivatives of the Riemann zeta function, I noted that, for any positive integer $n>1$, the following identity holds: $$\frac{\zeta'(n)}{\zeta^2(n)}=\sum_{x=1}^\infty \mu(x) \frac{\...
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1answer
50 views

Using Mobius inversion to determine coefficients.

Suppose we have a fixed positive integer $n$ and three functions $f:\mathbb N \longrightarrow \mathbb N$ and $g:\mathbb N\times \mathbb N\longrightarrow \mathbb N$ and $a:\mathbb N\rightarrow \mathbb ...
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moebius and powers. how do they behave?

Suppose $f(n)=q^n$, where $q$ is an integer $>2$. (In the original formulation, $q$ should have been a prime, but I think it doesn't really matter) Define the function $$ g(n) = \sum_{d|n} \mu\...
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1answer
214 views

On the Lambert series for the Möbius function

The Lambert series for the Möbius function is given by : $$\sum_{n=1}^{\infty}\mu(n)\frac{q^{n}}{1-q^{n}}=\sum_{n=1}^{\infty}\frac{\mu(n)}{q^{-n}-1}=q \;\;\;\;\;\; |q|<1$$ This follows from Möbius ...
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1answer
236 views

How many number of bracelets of length $n$ with black-white beads?

How many number of bracelets of length $n$ with black-white beads? I'm trying to find a formula for counting the number of such bracelets. What i've done so far is to think of the bracelets as binary ...
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2answers
190 views

Prove $ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$ where $\mu(n)$ is the Mobius function.

Given a positive integer $N$, show that $$ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$$ where $\mu(n)$ is the Mobius function. How do I approach this question? I guess a ...
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1answer
410 views

Circle inversion and the Pappus chain paradox

As is well-known, lines and circles are converted into lines and circles by circle inversion, or by any Möbius transformation for that matter. What bothers me is what happens in the Pappus's classical ...
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1answer
171 views

How do I prove the following equality using Mobius inversion.

$$\sum_{i=1}^n \frac{n}{gcd(i,n)} = \sum_{d|n} d\times\phi(d) $$ Where $\phi(d)$ is the Euler totient function of d. I want to prove it using Mobius Inversion formula. Please give a formal step by ...
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133 views

What is the use of Proposition 1 (Section 6) or Rota (1964)?

In the paper "On the Foundations of Combinatorial Theory I. Theory of Miibius Functions" published in 1964, Gian Carlo Rota laid down te means for the widespread use of the Mobius inversion formula in ...
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133 views

In the definition of the Mobius function and basic properties, can we change square-free to something else?

The Mobius function isolates square-free numbers ie numbers. Can we do this with other numbers with certain properties?
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2answers
827 views

Intuition for Möbius function on a poset

I am attempting to learn about Möbius inversion in the context of partial order theory. However, I'm hitting a bit of a mental block when it comes to understanding the Möbius function, and I'm looking ...