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Questions tagged [mobius-function]

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

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Reasoning about $\left(\left\lfloor\frac{2x}{i}\right\rfloor -2\left\lfloor\frac{x}{i}\right\rfloor\right)$

I am working on an alternative argument for Bertrand's Postulate that depends on the following argument. Please let me know if I made a mistake or if any point is unclear. Let: $p_k$ be the $k$th ...
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1answer
24 views

How to calculate the mobius function of a Poset using Hall's theorem

Hall's Theorem states that: $u(x,y) = C_0-C_1+C_2-C_3+...$ where $C_k$ is number of chains of length $k$ If $x\neq y$ then $C_0=0$ and $C_1=1$ But my question is why does $C_1$ have to equal ...
2
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0answers
40 views

Question about Dirichlet Series Related to Formula for $\frac{1}{e}$

This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens ...
1
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2answers
56 views

Show that there is always one integer $t$ with a least prime factor $> 5$ where $x < t \le x+6$

Let $p_k$ be the $k$th prime. Let $f_2(x) = \lfloor x\rfloor - \left\lfloor\dfrac{x}{2}\right\rfloor$ For $k > 1$, let: $f_{p_k}(x) = f_{p_{k-1}}(\lfloor x\rfloor) - f_{p_{k-1}}\left(\left\...
5
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1answer
51 views

Show $S(n) = \sum_{d=1}^n \mu(d) [\frac{n}{d^2}]$

Question: Prove that $$S(n) = \sum_{d=1}^n \mu(d) \left[ \frac{n}{d^2}\right],$$ where $\displaystyle \left[\frac{n}{d^2}\right]$ denotes the largest integer that does not exceed $\displaystyle \frac{...
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1answer
33 views

n-th root and Möbius function

Let $q$ be a prime power and $n$ a positive integer s.t. $\gcd(n, q) = 1$. Let $E(n)$ be the set of the complex $n$-th roots of unity. For every positive integer $d$ such that $d | n$ , let $$Q_d(x) ...
2
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1answer
38 views

$\sum_{d|n, d>0} (\sigma(d)/d)\mu(n/d))=1/n$

We want to show \begin{align} \sum_{d|n,\ d>0}(\sigma(d)/d)\cdot \mu(n/d) =1/n , \end{align} where $\sigma(m)$ denotes the sum of all positive divisors of $m$ and where $\mu$ is the Möbius ...
1
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1answer
61 views

What is the Dirichlet Transform of $a(n)=\frac{\mu(n)}{n^2}\,\log\left(\frac{2\,\pi}{n}\right)$?

This question is related to my previous question at the following link. Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$? This ...
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1answer
143 views

Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$?

This question is related to the following formula for Euler's constant $\gamma$ where $A$ is Glaisher's constant. (1) $\quad\gamma=12\,\log(A)-\frac{\pi^2}{6}\sum\limits_{n=1}^N\frac{\mu(n)}{n^2}\,\...
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0answers
50 views

Riemann Zeta function, perfect powers, and the Mobius function

I was toying around with the Riemann Zeta function recently and noticed that I could get to a particular representation (valid for $Re(s)>1$) in a couple of different odd ways. The first was by ...
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0answers
38 views

Show the followed equality, mobius function

Suppose $P$ is a finite poset, and $f: P \to \mathbb{C}$. Is it true that $$\sum_{x_1<x_2<...<x_k}{(f(x_1)-1)(f(x_2)-1)(f(x_3)-1)...(f(x_k)-1)} = \sum_{x_1<x_2<...<x_k}(-1)^k \mu(0,...
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1answer
69 views

Prove the sum of the Mobius function over monic polynomials of degree $n$ is $0$ if $n > 1$

Let $\mu(m)$ be the Möbius function on monic polynomials in $\mathbb{F}_q[x]$ ($q$ is power of prime) where $\mu(m) = 0$ if $m$ is not square-free and $\mu(m) = (-1)^k$ if $m$ is square-free and can ...
3
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1answer
76 views

The number of ways, $sf_2(n)$, to express an integer as the sum of two square-free integers.

It is well known that $$\sum_{n\leq x}\mid \mu(n) \mid \sim \frac{6}{\pi^2}x\left(1+o(1)\right) \text{, } x \to \infty$$ From here it follows that every sufficient large integer may be expressed as ...
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0answers
21 views

Name of poset operation involving Möbius function

If we have a finitary poset $P$ with a $\hat{0}$ (least element), and we want to compute the Möbius function for all elements $x$, as in $\mu(\hat{0},x)$, it wouldn't affect any of our computations to ...
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0answers
34 views

Upper bound for the reciprocal of a sum involving Möbius function

For $ n $ an integer greater than $ 6 $ , let $ Q(n)=\prod_{p\leq\sqrt{2n-3}}p $. Which upper bound in terms of $ n $ can we get for $ (\sum_{d\mid Q(n)}\frac{\mu(d)}{d})^{-1} $?
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1answer
50 views

How can one calculate the Möbius function $\mu(a_1,a_i)$ for all $i \in \{1, …, 10\}$ of this poset?

I've seen this partially ordered set in our combinatorics script and it says that it is obvious how to calculate the möbius function $\mu(a_1,a_i)$ for all $i \in \{1, ..., 10\}$. Here's the Hasse ...
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0answers
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Odd Mertens function

Let $M^*(n)$ be the "odd Mertens function", defined by $M^*(n) = \sum \mu(k)$ for odd $k$, $1 \le k \le n$. Let $r$ be an odd number. Since $\mu(r)$ is multiplicative, $\mu(2r) = -\mu(r)$ and $\mu(...
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1answer
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Looking for the name and properties of ${\varphi}_{2} (r, N) = \sum_{- N \le s, t \le N, (r, s, t) = 1} 1$ and $\sum_{d \mid r} \mu (d)/d^2$

I am counting the number of unique polynomial candidates for a fixed $r$ where $1 \le r \le N$ with $|s|, |t| \le N$ for naive height $N \ge r$. This sum is $${T}_{2} \left({r, N}\right) = \sum_{\...
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1answer
83 views

Do these arguments correctly predict $\sum_{r=1}^n \mu(r)$

Questions Is the below approach and proof correct? If I take $x \to 1- \epsilon$ and $n $ is of order $ \frac{1}{\epsilon^s}$ where $s \geq 2$ (so that the error $\to 0$) Then as $$ \sum_{r=1}^\...
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0answers
27 views

Simplifying an Expression for an Arithmetic Function

Question "Evaluate" or express $g_k(n)=\sum_{d\mid n,\,(d,k)=1}\mu(d)$ (where $k\in\mathbb{N}$ is fixed) in terms of elementary arithmetic functions. My attempt Using the fundamental property ...
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1answer
63 views

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

Let $k\in \Bbb{N}$ and $k>2$. Fix $k$, define $f_k(n)=\gcd(n,k)$ for $\forall\ n>1$. Let $\mu$ denote the Möbius function. Notice that $$\sum_{d\mid n}f_k(d)\mu\left(\frac{n}{d}\right)=0$$ for ...
2
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1answer
29 views

Möbius function - understanding of relations

I am trying to understand Möbius function from the wikipedia article (and also few others that I have come across so far). This function is defined in posets and so the relations in Special elements ...
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2answers
38 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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0answers
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Mobius Function over Euler Totient Function

The question is to prove: $\sum_{\phi(n)=k}\mu(n) = 0$ where $\phi(n)$ is the Euler totient function and $\mu(n)$ is the Mobius function. I have tried various approaches but nothing seems to be ...
3
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0answers
71 views

Calculating $\sum_{\substack{k|r \\ k \leq n}} \mu \left({ {k}}\right)$?

Background & Question I recently thought of a combinatoric method to get an interesting result: $$ \sum_{r=n+1}^{n!} \sum_{\substack{k|r \\ k \leq n}} \mu \left({ {k}}\right) = n! O(\frac{1}{\...
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0answers
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Questions related to the Riemann Xi function $\xi(s)$ and Jacobi theta functions $\vartheta_3(0,q)$

This question assumes the following definitions. (1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$ (2) $\quad f(x)=\sum\...
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2answers
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Interpretation of Coefficients of Expanded Cyclotomic Polynomials

Working out the following definition of the Cyclotomic Polynomial $$ {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right),} $$ you'll ...
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1answer
44 views

Is there an arithmetic function $\alpha(n)$ such that $\mu(n)=(\alpha*1)(n)$?

I've been looking into finding an arithmetic function $\alpha(n)$ for which its Dirichlet Convolution with the constant function $1$ is the Mobius Function, i.e.$$(\alpha(n)*1)=\mu(n)$$ I do not know ...
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Sum of a multiplicative arithmetic function

Prime factorization of $n$ is $\prod p_i^{e_i}$ Then radical of $n$ is defined as $\text{rad}(n)=\prod p_i$ Let $S(N) = \sum_{n=1}^{N}\text{rad}(n)$ I want to calculate $S(N)$ for very large value ...
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0answers
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Is $\sum^\infty_{n=1}\mu(n)z^n$ a lacunary function?

Let $\mu(n)$ be the mobius function. Then, is $$f(z)=\sum^\infty_{n=1}\mu(n)z^n$$ a lacunary function? Clearly, the series converges in the open unit disk. Since I have read from somewhere (maybe ...
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0answers
59 views

Asymptotic estimate for the sum $\sum_{n\leq X}\mu(n)\tau(n)$?

Just trying to figure out what would be the asymptotic relation for the expression $\sum_{n\leq X}\mu(n)\tau(n)$, where $\tau$ corresponds to the number of divisors function (often named $\sigma_0$ ...
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1answer
112 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
61 views

Questions on $f(x)=\sum\limits_{n=1}^{x}a(n)$ with an infinite number of positive integer zeros

This question is related to a class of functions that meet the following conditions. (1) $\quad f(x)=\sum\limits_{n=1}^{x}a(n)$ (2) $\quad f(x)=0$ for an infinite number of values of $x\in\mathbb{Z}^...
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1answer
34 views

Degree of a formal power series involving Mobius function

I am reading Enumerative Combinatorics by Richard Stanley, and I came across the following expression: $(1-x^n)^{\frac{-\mu(n)}{n}}$, where $\mu(n)$ is the usual Mobius function from number theory. ...
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0answers
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Using the Mobius function for a poset to solve for μ({1},{1,2,3,4}). and μ({1,2},{1,3,4}).

Consider the poset $(\mathcal{P}(S),\rho)$ where $S=\{1,2,3,4\}$ and $\forall A,B \in \mathcal{P}(S)$, $A\rho B$ if and only if $A\subseteq B$. Let $\mu: \mathcal{P}(S)\times \mathcal{P}(S)\...
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2answers
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How do we show that the sum is equal to $0$?

We have the arithmetic function $$f(n)=\sum_{d\mid n}\mu (d)\cdot d$$ I want to show that if $n$ is divisible by $p^2$ for some prime $p$ then $\displaystyle{\sum_{d\mid n}f(d)\mu \left (\frac{n}{d}\...
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1answer
43 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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1answer
43 views

Dubious step in 'order of mobius' proof which may lead to an interesting fact

Let $\mu(n)$ be the Moebius function, let $M(x)=\sum_{n\leq x} \mu(x)$ be the Mertens function and let $A(x)=\sum_{n\leq x}\tfrac{\mu(n)}{n}$ be the truncation of the Dirichlet series expansion of $1/\...
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0answers
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Upper bound of $ \sum_{n\leq x}f(n) $ where $ f(n)=\sum_{r=1}^{n-1}\mu(r)\mu(n-r) $

$Cx^2$ is a trivial bound by just counting the total number of terms in these sums. From here I have attempted to use $$ \sum_{n\leq x} \mid \mu(n) \mid =\frac{6}{\pi^2}x(1+o(1)) \text{, } x \to \...
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1answer
76 views

Landau's theorem with nth roots

Let $\omega(n)$ denote the number of distinct primes dividing $n$. Then the Mobius function $\mu(n)$ is defined by $\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree and $\mu(n) = 0$ otherwise. $\,$ ...
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0answers
58 views

Kind of a closed form for the Möbius function as the number of points on n-dimensional hyperboloids.

Through experimenting with the Mertens function sum I wrote in Wikipedia I found this formula if you like, for the Möbius function as a sum of the number of points on n-dimensional hyperboloids: $$\...
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0answers
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Expressing an integer as the sum of three of more square free numbers.

From the formula $$\sum_{n \leq x}\mid \mu(n) \mid=\frac{6}{\pi^2}x(1+o(1)) $$ one can show that every sufficiently large number can be written as the sum of two square-free numbers. The number of ...
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0answers
39 views

Behaviour of the sum $\sum_{r+s=n}\mu(r)\mu(s)$

I am interested in the large n behaviour of this sum. Every sufficiently large integer can be written as the sum of two square free numbers so this is a well defined question. I conjecture that this ...
6
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1answer
72 views

Oscillations of a Mobius sum

Define $ S(n) = \sum_{k=1}^n {\mu(k)\over k}$ . It is known but nontrivial that $S(n)$ approaches zero as $n$ approaches infinity. Here we are interested in the sign of $S(n)$ . Note that the ...
2
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1answer
77 views

Mobius function of the power set

I'm having difficulties understanding the derivation of the Mobius function for the power set, and would like to ask some questions. Does the equality "1" come about as the result of the induction ...
3
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2answers
50 views

Mobius function of product of primes

We have the arithmetic function $$f(n)=\sum_{d\mid n}\mu (d)\cdot d$$ I want to show that $f\left (p_1^{e_1}\cdots p_k^{e_k}\right )=(-1)^k\cdot (p_1-1)\cdots (p_k-1)$. We have that $d$ is of the ...
2
votes
1answer
57 views

Ordinary generating function for $\mu^2(x)$

I'm trying to find the ordinary generating function $f(n)$ such that $$f(n) = \sum_{k=0}^\infty\mu^2(n)\, x^k$$ I have a nasty looking answer that involves Hadamard products, and I was hoping there ...
4
votes
1answer
76 views

Do you know a generalization of a formula about cyclotomic polynomial?

Let n be a natural number. Then n-th cyclotomic polynomial is defined as follows: $$\Phi_{n}(x)=\prod_{k\in\mathbb{N}_{<k},(n,k)=1}(x-\zeta^k)$$ where $\mathbb{N}_{<k}$ means the set of natural ...
0
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1answer
61 views

Dirichlet series for 1/ζ(s)

Prove that for Re(s)>1 $$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$$ Where $\mu(n)$ is the Möbius function defined by: $\mu(n)=1, \mbox{if }n=1$ $\mu(n)=(-1)^k, \mbox{if }n=p_1,p_2,....
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1answer
186 views

Dirichlet Series of Absolute value of Mobius Function equals Ratio of Riemann Zeta

I would like to prove this using Euler products: $$\frac {\zeta(s)}{\zeta(2s)} = \sum_{n=1}^{\infty}\frac {\lvert \mu(n) \rvert}{n^s}$$ I have gotten here, but don't know if this is a correct ...