Questions tagged [mobius-inversion]

For questions related to Möbius inversion and its applications.

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24 views

Relation between $f$ and $g$ satisfying $\sum_{r}^N f(x^r) = g(x)$ without mobius inversion?

Let's say for a particular function I have $f$: $$ f(x) + f(x^2)+ f(x^3)+ \dots +f(x^N) = g(x) $$ We start by considering the integral: $$ I_f = \int_0^{b} f (e^{-\frac{1}{x}}) dx $$ Using asymptotics ...
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2answers
55 views

Prove inversion formula involving binomial coefficients

Let's say that we have such an equation: $$f_k = \sum_{i=0}^{k}{k \choose i}g_i$$ Prove that in that case we can express $g_k$ like this: $$g_k = \sum_{i=0}^{k}(-1)^{k-i}{k \choose i}f_i$$ In order to ...
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1answer
73 views

Is $ \ln \delta \int_0^1 \frac{f(z)}{z(\ln z)^2} dz \sim \sum_{r=1}^\infty \mu(r) f(\delta^{1/r}) $?

Background I think I can show something interesting: $$ \ln \delta \int_0^1 \frac{f(z)}{z(\ln z)^2} dz \sim \sum_{r=1}^\infty \mu(r) f(\delta^{1/r}) $$ where $\delta \to 0$ and $\int_0^1 \frac{f(z)}{...
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100 views

Understanding the method of using inversion to solve a problem on touching circles

Need to prove: (1) The points of contact of the chain of circles $ (C_o, C_1, C_2..C_n)$ has the locus of a semi-circle (2) if radius of $C_n$ is $ r_n$, then height above L of the center of $ r_n$ is ...
4
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2answers
59 views

Intuition behind inversions of triangles in circles leading to similar triangles

In here the inversion of triangle $qab$ produces triangle $q \tilde{a} \tilde{b}$ but I find it a bit weird that $\angle qab = \angle q \tilde{b} \tilde{a}$ and $\angle q ab = \angle q \tilde{a} \...
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82 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
69 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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52 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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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
43 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
65 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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31 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
38 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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40 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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1answer
61 views

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
53 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
151 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
35 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
78 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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41 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
84 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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65 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
58 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
92 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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32 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
88 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
139 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
201 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
67 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
50 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
41 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
89 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
70 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
357 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
93 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
77 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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48 views

$f(n) = \Sigma_{d|n} \mu(n/d)F(d)$

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
223 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
290 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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42 views

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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42 views

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
243 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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51 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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25 views

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
225 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
254 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
228 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 ...