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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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Rational number sets which grow in cardinality and a related sequence of numbers.

Define the collection of sets $\left\{Q_n\right\}$ as follows: $$Q_1 = \{0\};\quad Q_2 = \left\{\frac{1}{2}\right\} \cup Q_1; \quad Q_3 = \left\{\frac{1}{3}, \frac{2}{3}\right\} \cup Q_2;$$ $$\left\{\...
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The relation between the number of square-free divisors of n and the Möbius $\mu$- function.

I was solving this problem: Let $S(n)$ denote the number of squarefree divisors of $n$. Establish that $$ S(n) = \sum_{d \mid n} \lvert \mu(d) \rvert = 2^{\omega(n)},$$ where $\omega(n)$ is ...
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Time of calculating Mertens function.

There is Mertens function: $$M(x)=\sum_{n \le x}\mu(n)$$ How to calculate time of computing value of $M(N)$ for particular $N$? I would like for example find that time using this formula: $$M(x)=1-...
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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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Trying to count the number of integers $x \le n$ where gcd$\left(x(x+2),30\right)=1$ using the möbius function

Let: $x>0, n >0$ be integers gcd$(s,t)$ be the greatest common divisor of $s$ and $t$ $\mu(x)$ be the möbius function For $x \le n$ and gcd$(x,30)=1$, the count is: $$\sum_{i | 30}\left\...
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Summing $M(n)$ and more

$M(n)$ is Merten's function which is the sum of the Mobius function, $\mu(n).$ $$M(n)=\sum \mu(n).$$ Define the following functions: $$ \Phi(n)= \sum M(n) $$ $$ \Psi(n)= \sum \Phi(n) $$ $$ \Omega(...
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A summation of the multiplication of reflected Mobius functions and their behavior for different values of $k$

$$ \Psi_k(N)=\sum_{n=1}^{N} \mu(n)\mu(k-n)$$ where $\mu(n)$ is the Mobius function. This function is interesting to me because for the case of $k=N$ it has the symmetric property of being odd with ...
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Reasoning about remainders and the Möbius function

This one seems counter intuitive to me but I am not seeing a mistake in my reasoning. Please let me know if you find one. Let: $x > 0$ be an integer $\mu(x)$ be the möbius function $x\#$ be the ...
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Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
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Counting the number of integers with their least prime factor greater than $x$ between $ax$ and $ax+x$

Let: $x \ge 2, a \ge 1$ be integers. $x\#$ be the primorial for $x$ $\mu(i)$ be the möbius function. $\text{lpf}(x)$ be the least prime factor of $x$. $p_k$ be the $k$th prime which is the highest ...
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Prove $F^* = \mu * F$

Let $f: \mathbb{Q} \cap [0,1] \to K$ and set $F(n) = \sum_{k = 1}^n f(\frac k n)$, $F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$. Show that $F^* = \mu * F$ where $*$ is the Dirichlet product....
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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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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 ...
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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 ...
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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\...
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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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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) ...
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$\sum_{d|n, d>0} (\sigma(d)/d)\mu(n/d))=1/n$

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

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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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 ...
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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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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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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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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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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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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
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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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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
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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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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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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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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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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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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 \...