Questions tagged [mobius-function]
Questions on the Möbius function μ(n), an arithmetic function used in number theory.
2
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2answers
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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 ...
1
vote
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
votes
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
vote
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
votes
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{...
0
votes
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
$$Q_d(x) ...
2
votes
1answer
38 views
$\sum_{d|n, d>0} (\sigma(d)/d)\mu(n/d))=1/n$
We want to show
\begin{align}
\sum_{d|n,\ d>0}(\sigma(d)/d)\cdot \mu(n/d) =1/n ,
\end{align}
where $\sigma(m)$ denotes the sum of all positive divisors of $m$ and where $\mu$ is the Möbius ...
1
vote
1answer
61 views
What is the Dirichlet Transform of $a(n)=\frac{\mu(n)}{n^2}\,\log\left(\frac{2\,\pi}{n}\right)$?
This question is related to my previous question at the following link.
Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$?
This ...
4
votes
1answer
143 views
Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$?
This question is related to the following formula for Euler's constant $\gamma$ where $A$ is Glaisher's constant.
(1) $\quad\gamma=12\,\log(A)-\frac{\pi^2}{6}\sum\limits_{n=1}^N\frac{\mu(n)}{n^2}\,\...
1
vote
0answers
50 views
Riemann Zeta function, perfect powers, and the Mobius function
I was toying around with the Riemann Zeta function recently and noticed that I could get to a particular representation (valid for $Re(s)>1$) in a couple of different odd ways.
The first was by ...
1
vote
0answers
38 views
Show the followed equality, mobius function
Suppose $P$ is a finite poset, and $f: P \to \mathbb{C}$. Is it true that
$$\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,...
1
vote
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
votes
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 ...
1
vote
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 ...
0
votes
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} $?
1
vote
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 ...
1
vote
0answers
31 views
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(...
0
votes
1answer
14 views
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 $${T}_{2} \left({r, N}\right) = \sum_{\...
3
votes
1answer
83 views
Do these arguments correctly predict $\sum_{r=1}^n \mu(r)$
Questions
Is the below approach and proof correct? If I take $x \to 1- \epsilon$ and $n $ is of order $ \frac{1}{\epsilon^s}$ where $s \geq 2$ (so that the error $\to 0$)
Then as $$ \sum_{r=1}^\...
0
votes
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 ...
4
votes
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
votes
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. ...
2
votes
0answers
45 views
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
votes
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:
$$ \sum_{r=n+1}^{n!} \sum_{\substack{k|r \\ k \leq n}} \mu \left({ {k}}\right) = n! O(\frac{1}{\...
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0answers
69 views
Questions related to the Riemann Xi function $\xi(s)$ and Jacobi theta functions $\vartheta_3(0,q)$
This question assumes the following definitions.
(1) $\quad\psi(x)=\sum\limits_{n=1}^\infty e^{-\pi\,n^2\,x}=\frac{1}{2} \left(\vartheta_3\left(0,e^{-\pi\,x}\right)-1\right)$
(2) $\quad f(x)=\sum\...
0
votes
2answers
80 views
Interpretation of Coefficients of Expanded Cyclotomic Polynomials
Working out the following definition of the Cyclotomic Polynomial
$$ {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right),}
$$
you'll ...
0
votes
1answer
44 views
Is there an arithmetic function $\alpha(n)$ such that $\mu(n)=(\alpha*1)(n)$?
I've been looking into finding an arithmetic function $\alpha(n)$ for which its Dirichlet Convolution with the constant function $1$ is the Mobius Function, i.e.$$(\alpha(n)*1)=\mu(n)$$
I do not know ...
0
votes
0answers
52 views
Sum of a multiplicative arithmetic function
Prime factorization of $n$ is $\prod p_i^{e_i}$
Then radical of $n$ is defined as $\text{rad}(n)=\prod p_i$
Let $S(N) = \sum_{n=1}^{N}\text{rad}(n)$
I want to calculate $S(N)$ for very large value ...
0
votes
0answers
22 views
Is $\sum^\infty_{n=1}\mu(n)z^n$ a lacunary function?
Let $\mu(n)$ be the mobius function.
Then, is
$$f(z)=\sum^\infty_{n=1}\mu(n)z^n$$
a lacunary function?
Clearly, the series converges in the open unit disk.
Since I have read from somewhere (maybe ...
2
votes
0answers
59 views
Asymptotic estimate for the sum $\sum_{n\leq X}\mu(n)\tau(n)$?
Just trying to figure out what would be the asymptotic relation for the expression
$\sum_{n\leq X}\mu(n)\tau(n)$,
where $\tau$ corresponds to the number of divisors function (often named $\sigma_0$ ...
0
votes
1answer
112 views
Identity involving the Möbius function and the first derivative of the Riemann zeta function
Working on the derivatives of the Riemann zeta function, I noted that, for any positive integer $n>1$, the following identity holds:
$$\frac{\zeta'(n)}{\zeta^2(n)}=\sum_{x=1}^\infty \mu(x) \frac{\...
0
votes
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}^...
2
votes
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
33 views
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)\...
1
vote
2answers
74 views
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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votes
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 ...
1
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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/\...
1
vote
0answers
38 views
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 $$ \sum_{n\leq x} \mid \mu(n) \mid =\frac{6}{\pi^2}x(1+o(1)) \text{, } x \to \...
2
votes
1answer
76 views
Landau's theorem with nth roots
Let $\omega(n)$ denote the number of distinct primes dividing $n$. Then the Mobius function $\mu(n)$ is defined by $\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree and $\mu(n) = 0$ otherwise. $\,$ ...
3
votes
0answers
58 views
Kind of a closed form for the Möbius function as the number of points on n-dimensional hyperboloids.
Through experimenting with the Mertens function sum I wrote in Wikipedia I found this formula if you like, for the Möbius function as a sum of the number of points on n-dimensional hyperboloids:
$$\...
1
vote
0answers
39 views
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 ...
2
votes
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
votes
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
votes
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
votes
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
votes
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,....
1
vote
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 ...
