Questions tagged [mobius-function]
Questions on the Möbius function μ(n), an arithmetic function used in number theory.
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Exploring Properties of the Mertens Function: A Potential Theorem
Dear members of the Math Exchange community,
I would like to share with you an interesting conjecture involving the Mertens function:
Theorem- $M(2^{m})<m$
Below, I have outlined a proof of this ...
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a question on $\sum _{n=1}^{\infty } \frac{\mu (n)}{n}$
We know this sum is $0$. But as I expanded it, I couldn't help but notice this is the inclusion exclusion principle in action.
$\sum_{n=1}^{\infty } \frac{\mu (n)}{n} = 1 - \sum\limits_{i = 1}^{\infty}...
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Freshman's dream and the commutativity of the square root of the Möbius function over the divisors.
Let the infinite matrix $A$ be:
$$A(n,k)=\left[ k \mid n \right] \left(\frac{\sqrt{k \, \mu(k)}}{n^s}\right)$$
where $n=1,2,3,4,5,...$ and $k=1,2,3,4,5,...$
Multiply $A$ with its transpose $A^{\mathsf{...
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Möbius function for graphs??
I'm reading this post and I'm getting a little confused. I am trying to find a useful notion of the Mobius function for directed graphs and have had little success in my search. I don't know much ...
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Moebius function and expansion in sequence $A280194$
For my research interest, I came accross with OEIS sequence A280194
In particular I am interested how is obtained the value $36203$ in the sequence
Expansion of:
$\frac{1}{1-\sum_{k>=1}{\mu(k)^2\...
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1
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Automorphisms on upper half plane
I am supposed to prove that: \begin{align} Aut(\mathbb{H})= \left\{ T:\mathbb{H}\to \mathbb{H}, T(z)=\frac{az+b}{cz+d} \vert a,b,c,d\in\mathbb{R}; ad-bc>0 \right\} \end{align}.
where $\mathbb{H}$ ...
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Alternating Dirichlet series involving the Möbius function.
It is well known that:
$$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$
with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function.
Numerical ...
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Proof of the dual Möbius Inversion Formula
The Dual Möbius Inversion Formula: Let $\mathcal D$ be a divisor closed set of natural numbers (i.e., if $d\in \mathcal D$ and $c \mid d$, then $c\in \mathcal D$.) Let $f$ and $g$ be two complex-...
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Show that $\Phi(n) = \phi(1) +\phi(2) + \dots + \phi(n) =\frac{n^2}{2 \zeta(2)}+O(n \log n)$
Set $\Phi(n) = \phi(1) + \phi(2) + \cdots + \phi(n)$.
I want to understand why
$$
\Phi(n) = \frac{n^2}{2\zeta(2)} + O(n\log n).
$$
I am going to show the proof that I have, and I would be very ...
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Invariance of the Möbius function under order isomorphism
Let $(X,\leq)$ and $(Y,\leq)$ be locally finite partially ordered sets
with Möbius functions $\mu_X$ respectively $\mu_Y$,
and let $\varphi:X\to Y$ be an order isomorphism.
I want to show that
\begin{...
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Summing a series with number-theoretic coefficients
Is there a simpler form to $f(x)=\sum_{r=1}^{\infty}\left(\sum_{d|r}d\mu(d)\right)x^r/r$ where $\mu(d)$ is the Mobius function and $\sum_{d|r}$ is a sum over the divisors $d$ of $r$?
I searched the ...
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Calculate $ \sum_{k=1}^{n} k\cdot \mu(k) $
Problem:
Given $n$, Calculate $ \sum_{k=1}^{n} k\cdot \mu(k) $
This is the oeis series.
My Thoughts:
I need a sublinear algo, possibly something of the order of $n^{3/4}$ or $n^{2/3}$ time. Any ...
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Is there a link between the partition function of $n$ and the Möbius function?
Conjecture: the partitions of $n$ and the Möbius function are related through the Möbius inversion formula.
The Möbius function is denoted by $μ(n)$ and is defined as follows:
$μ(n) = 1$ if $n = 1$
$μ(...
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Two poset properties: are they related?
A bunch of infinite posets $P$ with $\hat 0$ have the following property
For every $x\in P$, the principal filter $\{ y\in P : y\ge x\}$ is isomorphic as a poset to $P$ itself.
Examples include ${\...
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What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by
$$
A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p
$$
is this serie calculated ...
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New $ \overset {s}{\underset{k=x}{ \lower 3pt { \LARGE\Xi}}} $ operator and Möbius function
At the beginning I will explain some concepts and at the end I will ask the specific question.
Let's consider some operator $ \overset {s}{\underset{k=x}{ \lower 3pt {
\LARGE\Xi}}} $ on function $f ...
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Characterization of Möbius-monotonicity
We say that an arithmetic function $f:\mathbb{Z}^+\to\mathbb{C}$ is Möbius-monotone if $\forall n\geq1:(\mu*f)(n)\geq0$ (where $*$ denotes de Dirichlet convolution), i.e. if there exists a non-...
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Prove the sum of nth complex primitve roots of unity is $\mu(n)$ [duplicate]
I am trying to prove the following statement,
Let ω be a complex nth root of unity. We say that ω
is a primitive nth root if $\omega^m\neq1$ for any positive $m < n$. Prove that for all
$n\in\...
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Product version of Möbius Inversion formula
I am currently learning about Möbius Inversion. In Wikipedia , the sum version is proved using convolutions which I can follow along with. However, it also lists a product version of this formula:
$g(...
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Definition of the Möbius function from Vinogradov's book
Ivan Vinogradov in the book "Elements of Higher Mathematics" (page 357) gives the following definition of the Möbius function:
The Möbius function is a multiplicative function defined by ...
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Proving that $\sum_{k\le x}\frac{k}{\varphi(k)}=Ax+O(\log x)$
I'm hoping to show that $\sum_{k\le x}\frac{k}{\varphi(k)}=Ax+O(\log x)$, for some positive constant $A$ but I keep getting stuck.
I know that $\sum_{k\le x}\frac{1}{\varphi(k)} = O(\log x)$ and $\...
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Prove this generalization of Möbius inversion formula
Generalizations list this one as a generalization of Möbius inversion formula-
Suppose $F(x)$ and $G(x)$ are complex-valued functions defined on the interval $[1, ∞)$ such that
$$G(x)=\sum_{1\le n\le ...
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Möbius inversion and Möbius function as sum of cosines
Let $\mu (n)$ be the Möbius function. I want to prove the following formula:
$$\mu (n)=\sum_{\substack{1\leq k \leq n\\ (k,n)=1}}\cos \frac{2k\pi}{n}.$$
Let $F(n)$ be the right hand side, then by ...
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Perron's formula application - zeros of $L(s,\chi )$
After applying Perron's formula I have a complex integral involving something like
\[ \sum _{n=1}^\infty \frac {\mu (n)}{n^s}e\left (\frac {an}{q}\right ).\]
As usual ("usual" meaning e.g. ...
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$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\frac xd\right\rfloor$ and $\sum_{d|n}\varphi\left(\frac xd,\frac nd\right)=\lfloor x\rfloor$
If $x$ is real, $x\ge 1$, let $\varphi(x,n)$ denote the number of positive integers less than or equal to $x$ that are relatively prime to $n$. Prove that
$$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\...
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Need help with finding Dirichlet generating function
I am unable to find the following generating function:
$$\tag{1}{\sum _{n=1}^{\infty } \frac{2^{-k_n} \mu \big(\frac{n}{2^{k_n}}\big)}{n^s}}$$
$\mu$ is Möbius function, $k_n$ is the highest integer ...
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Product form of Mobius Inversion formula: $g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$
Product form of the Möbius inversion formula: If $f(n)>0$ for all $n$ and if $a(n)$ is real, $a(1)\neq 0$, prove that
$$g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$$
where $b=a^...
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Prove that $\sum_{d|n}\mu(d)\log^m d=0$
Prove that
$$\sum_{d|n}\mu(d)\log^m d=0$$
if $m\ge 1$ and $n$ has more than $m$ distinct prime factors.
I tried using Induction and kind of succeeded in the sense that if we write down the case $m=2$ ...
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Count pairwise coprime triples such that the maximum number of the triple is not greater than N
Problem Statement:
Given N you are to count the number of pairwise coprime triples which satisfy $1≤a,b,c≤N$.
Example:
For example N=3,
valid triples are (1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,1,3),(1,3,1)...
user1068604
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The image of a Möbius transformation
I have the following problem it seems easy but I couldn't find the answer!
For the following $m(z)$ $=$ $\frac {z - i}{iz-1 }$ show that $m$($\mathbb{H}$) $=$ $\mathbb{D}$
I tried to show that |$m(z)...
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Calculate values of Möbius function
Let $(P,\leq)$ be a partial order with $P = \lbrace 0,1,2,3,4 \rbrace$ such that $0 \leq 1 \leq 4$,$0 \leq 2 \leq 4$,$0 \leq 3 \leq 4$. I want to calculate the values of the mobius function $\mu : P \...
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Show that $\sum \frac{\mu(n) \ln(n)}{n}=-1$
I am thinking of interpreting this as $-\frac{d}{ds}(\frac{1}{\zeta(s)})$ evaluated at $s=1$, or connecting it with $\Lambda$, but not exactly sure how to.
Any reference/textbook to look at for such ...
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Using Mobius Inversion Formula
We are given the following:
$f(n)=\prod_{d|n}g(d)$
and asked to show:
$g(n)=\prod_{d|n} f(d)^{\mu(\frac{n}{d})}$
The hint given says to use logarithms
Here's what I tried doing:
$log(f(n))=\prod_{d|n}...
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$\lim_{s \to 1^+} 1/\zeta(s) = 0$ obvious or not?
I read the statement that
$$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s} \textrm{ for } \Re(s) > 1 \qquad (*)$$
In fact I can guess what the proof is: just expand both $\zeta$ and the ...
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Analytic continuation of this function to $|z|>1$?
Consider the following function:
$$\sum_{n=1}^\infty \sum_{s | n} s \mu(s) z^{n^2},$$
where $\mu(s)$ is the Mobius function. This converges for $|z|<1$. Does there exist an analytic continuation to ...
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Question on cyclotomic polynomials and the Möbius function
I'm doing an exercise on the Möbius function $\mu$. I've seen this equation but I don't understand it.
\begin{align*}
\mathrm{X}^{n}-1&= \prod_{d \mid n} \phi_d(X) \\
&=\prod_{d \mid n} \left(\...
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Find asymptotic formula for $\sum\limits_{n\le x} \frac{1}{\phi(n)}$ where $\phi$ is Euler's Phi function [duplicate]
I know (and proved) an identity
$$\frac{1}{\phi(n)}=\frac{1}{n}\sum\limits_{d|n}\frac{\mu(d)^2}{\phi(d)}$$
Using this I got-
$\displaystyle{\sum\limits_{n\le x} \frac{1}{\phi(n)}}$
$\displaystyle{=\...
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Show that $\sum\limits_{n\leq x} \varphi(n)=\frac{1}{2}\sum\limits_{n\leq x}\mu(n)[\frac{x}{n}]^2+\frac{1}{2}$.
I am a graduate student of Mathematics.I am now studying analytic number theory from Apostol's book.In the exercise $3$ there is a question which is as follows:
If $x\geq1$,Show that $\sum\limits_{n\...
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Interpreting a potential bias of the Mertens function
In this question of some years ago on MO, the presence of a negative bias for the Mertens function was hypothesized.
A key point for such a problem is how the bias is defined. For example, if we focus ...
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What is the essence behind representing Möbius transformations as matrices?
I know that Möbius transformations generally maps lines/circles to line/circles using a function $f\left( z \right) = \frac{{az + b}}{{cz + d}}$ defined over $\mathbb C$. However, what I do not ...
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Divisor sum property of Euler phi function with Mobius inversion
I have the following formula of the Mobius inversion:
$$g(n) = \sum_{d|n}f(d) \iff f(n) = \sum_{d|n}g(\frac{n}{d})\mu(d)$$
The euler phi function has a divisor sum property: $\sum_{d|n}\phi(\frac{n}{d}...
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Prove that $\mu(n)\mu(n+1)\mu(n+2)\mu(n+3)=0$ for $n\ge 1$
I got this question that asks
Prove that
$$\mu(n)\mu(n+1)\mu(n+2)\mu(n+3)=0$$
where $\mu$ is the Mobius function.
So, basically, we need to prove that out of every four consecutive integers, atleast ...
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Why is $\sum_{d\mid 100} \sigma_2{(100)} \cdot \mu \left(\frac{100}{d}\right) = 100^2$?
I am working on a problem, but I am not sure why I get the wrong answer. The question asks, what is, $$\sum_{d\mid n} \sigma_2(d)\cdot \mu \left(\frac{n}{d}\right) \tag{ for n=100}$$
In the previous ...
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Prove $\sum_{d | n} \mu(d) (\log(d))^2=0$ [duplicate]
If $n$ is a positive integer with more than 2 distinct prime factors, how to prove that $\sum_{d | n} \mu(d) (\log(d))^2=0$?
I struggle on how to continue from this.
Suppose $n=p_1 p_2 ... p_r$, ...
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A divisor sum involving Moebius function and Jordan's totient function
I am trying to prove the following claim:
Let $\mu(n)$ be the Moebius function and let $J_k(n)$ be the Jordan's totient function. Then,
$$\displaystyle\sum_{d \mid n} \frac{\mu^2(d)}{J(k,d)}=\frac{n^...
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A divisor sum involving generalized Moebius function
I am trying to prove the following claim:
Let $n$ be a squarefree natural number. Denote by $\mu_k$ the generalized Moebius function: $\mu_k=\underbrace{\mu \ast \ldots \ast \mu}_{k}$ where $\ast$ is ...
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Variant of Möbius inversion: $b(n) = \sum_{d^2 \mid n} a(n/d^2) d^\alpha$
I'm trying to understand a step in a classic paper of Rankin. In Rankin's paper Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithmetical functions, he defines
$$ b(n) := \...
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A question related to behavior of mobius function $\mu(d)$
This is a continuation of a previous question.
Consider the sum $\sum_{d|P(z),d≤x}μ(d)\sum_{n≤x,d|n}1$, which is clearly equal to $\sum_{d|P(z),d≤x}μ(d)×(x/d+O(1))$. A text I'm reading claims this is ...
user775699
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Solve the equation with respect to $k_1,k_2\in \mathbb{Z}_{+}$
I am struggling with solving the following equation for positive integers $k_1$ and $k_2$ in terms of $n\in \mathbb{Z}_+$ and $i,j\in \mathbb{Z}_+$:
$$n-1=\sum_{i\le k_1,j\le k_2}\sum_{\text{gcd}(i,j)=...
user931598
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For $a(n)=\sum\limits_{d|n}\mu(d)\ \mu\left(\frac{n}{d}\right)$, does $\underset{x\to\infty}{\text{lim}}\left(\sum_{n=1}^x\frac{a(n)}{n}\right)=0$?
Consider the function $a(n)$ defined in formula (1) below, the related summatory functions defined in formulas (2) to (5) below, and their relationships with the Riemann zeta function $\zeta(s)$ ...
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