Questions tagged [mobius-function]

Questions on the Möbius function μ(n), an arithmetic function used in number theory.

The Möbius function $\mu(n)$ is defined for positive integers $n$ using their prime factorization. We have $\mu(1)=1$ and if $n>2$, $n=p_1^{a_1}\cdots p_r^{a_r}$, then $$\mu(n)=\begin{cases} 0 &\text{if }n\text{ is not square-free, that is, }a_k>1\text{ for some }k\\ (-1)^r &\text{otherwise.} \end{cases}$$

The Möbius function is multiplicative: if $m$ and $n$ are relatively prime, then

$$\mu(mn) = \mu(m) \mu(n).$$

One particularly important use of the Möbius function is the Möbius-inversion formula, which states that $\mu$ is the Dirichlet inverse of the constant function $1$.

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Coefficient of $n$th cyclotomic polynomial equals $-\mu(n)$

For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that $$\varphi(\Phi_n)=-\mu(n)$$ where $\mu(n)$ is the Möbius…
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Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

I'm trying to work through Ireland and Rosen's A Classical Introduction to Modern Number Theory as I've heard good things about it. This is Exercise 12 from Chapter 2. Here $\mu$ is the Moebius function, and $\phi$ the totient function. Find…
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Prove sum of primitive roots congruent to $\mu(p-1) \pmod{p}$

Suppose that $p$ is a prime. Prove that the sum of the primitive roots modulo $p$ is congruent to $\mu(p − 1) \pmod{p}$.
Johan
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The Möbius function is the sum of the primitive $n$th roots of unity.

Did you know that the Möbius function $\mu$ is the sum of the primitive nth roots of unity? I want to know about meaning of this. This statement is expressed as, $$\mu(n) = \sum_{\substack{k=1 \\ (k,n)=1}}^n \exp\left(\frac{2\pi ik}{n}\right).$$
J.U.math
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A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

Let $\mu$ denote the Möbius function whereby $$\mu(k) = \begin{cases} 0 & \text{if $k$ has one or more repeated prime factors} \\ 1 & \text{if $k=1$} \\ (-1)^j & \text{if $k$ is a product of $j$ distinct primes}\end{cases}$$…
John M. Campbell
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Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $

I am trying to show $$\sum_{d \leq x} \mu(d)\left\lfloor \frac{x}{d} \right\rfloor = 1 \;\;\;\; \forall \; x \in \mathbb{R}, \; x \geq 1 $$ I know that the sum over the divisors $d$ of $n$ is zero if $n \neq 1$. So we can rule out integers that are…
Tyler Hilton
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Are there infinitely many natural numbers $n$ such that $\mu(n)=\mu(n+1)=\pm 1$?

A while ago, I answered this question here on StackExchange which asks if for any given integer $k$, whether there exists infinitely many natural numbers $n$ such that $$ \mu(n+1)=\mu(n+2)=\mu(n+3)=\cdots=\mu(n+k) $$ This is indeed the case, and can…
Dylan
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How to prove that $n\sum\limits_{d\mid n}\frac{|\mu(d)|}{d}=\sum\limits_{d^2\mid n}\mu(d)\sigma\left(\frac{n}{d^2}\right)$?

This is problem 11 part b in chapter 3 of Tom M. Apostol's "Introduction to Analytic Number Theory". A variation on Euler's totient function is defined as $$\varphi_1(n) = n \sum_{d \mid n} \frac{|\mu(d)|}{d}$$ The question asks to show…
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Möbius function from random number sequence

Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}. Then plug it into the first column in a matrix that has the…
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Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = a_0x^n+a_1x^{n-1}+...+a_n$ ... Where the coefficients are the elements of…
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Squarefree totient sum

Does anybody have a reference/proof for the asymptotic growth rate of $$A(x) = \!\!\!\!\!\!\sum_{\substack{n \leqslant x \\ n \ \text{squarefree}}} \!\!\!\!\!\! \varphi(n)$$ as $x \to \infty$? Here $\varphi(n)$ denotes Euler's totient function.…
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What is the inverse of $\left[ \sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor \right]_{n \times n}$?

For $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ integer matrix whereby the $(i, j)$-entry of $M_{n}$ is equal to $\sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor$, for all indices $i$ and $j$. For example, we have that: $$ M_{5} =…
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On calculations for $\prod_{k=2}^\infty \left( 1+\frac{\mu(k)}{k^3} x\right)$, where $\mu(n)$ is the Möbius function

I define for some set of real numbers $x\in S$ (see that it is my Question 1.) the domain of the function $$f(x)=\prod_{k=2}^\infty \left( 1+\frac{\mu(k)}{k^3} x\right) ,$$ where $\mu(k)$ is the Möbius function. Question 1. I am interesting in this…
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A good estimation of the inverse $f^{-1}(x)$ of $-\sum_{n=2}^\infty\frac{\mu(n)}{n}x^n$, over the unit inverval if it has mathematical meaning

In my home I am trying to get the thesis for a specialization for my example* from a statement published related to functions that satisfy certain properties. I've considered the function* …
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Prove that $\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$

In my notes: $$\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$$ where $\mu(d)$ is the Möbius function and $\sigma(d)$ is the sum of all positive divisors of $d$. And I have no idea how they got the expression on the right hand side.…
Chan
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