Questions tagged [mobius-function]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
5 views

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 ...
renato de lima's user avatar
0 votes
0 answers
53 views

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}...
sku's user avatar
  • 2,433
1 vote
1 answer
69 views

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{...
Mats Granvik's user avatar
  • 7,286
0 votes
0 answers
46 views

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 ...
joe's user avatar
  • 129
1 vote
1 answer
63 views

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\...
Enzo Creti's user avatar
2 votes
1 answer
62 views

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 $\mathbb{H}$ ...
MilesDefis's user avatar
3 votes
1 answer
124 views

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 ...
Agno's user avatar
  • 3,007
1 vote
1 answer
114 views

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-...
stoic-santiago's user avatar
6 votes
1 answer
206 views

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 ...
Oswald Diaz's user avatar
3 votes
0 answers
26 views

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{...
Bart's user avatar
  • 964
1 vote
0 answers
61 views

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 ...
Valentine Michael Smith's user avatar
4 votes
1 answer
164 views

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 ...
sibillalazzerini's user avatar
0 votes
0 answers
32 views

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(...
Craw Craw's user avatar
5 votes
1 answer
68 views

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 ${\...
marcelgoh's user avatar
  • 1,776
0 votes
1 answer
42 views

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 ...
Es-said En-naoui's user avatar
1 vote
0 answers
120 views

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 ...
Wreior's user avatar
  • 388
1 vote
0 answers
42 views

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-...
Danka Makabre's user avatar
0 votes
0 answers
27 views

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\...
Mathsbot69's user avatar
-1 votes
1 answer
60 views

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(...
Meow's user avatar
  • 152
3 votes
2 answers
111 views

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 ...
Aragorn's user avatar
  • 75
1 vote
1 answer
37 views

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 $\...
James2390's user avatar
1 vote
1 answer
165 views

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 ...
Sayan Dutta's user avatar
  • 8,122
3 votes
6 answers
309 views

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 ...
Ishigami's user avatar
  • 1,535
0 votes
0 answers
94 views

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. ...
tomos's user avatar
  • 1,528
0 votes
2 answers
128 views

$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\frac xd\right\rfloor$ 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\...
Sayan Dutta's user avatar
  • 8,122
0 votes
1 answer
87 views

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 ...
azerbajdzan's user avatar
1 vote
1 answer
301 views

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^...
Sayan Dutta's user avatar
  • 8,122
5 votes
1 answer
165 views

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 $m=2$ ...
Sayan Dutta's user avatar
  • 8,122
6 votes
1 answer
165 views

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)...
user avatar
-1 votes
2 answers
52 views

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)...
Ola alfares's user avatar
1 vote
0 answers
52 views

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 \...
Orb's user avatar
  • 941
0 votes
0 answers
54 views

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 ...
flypig's user avatar
  • 117
2 votes
0 answers
83 views

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}...
user8083's user avatar
  • 131
5 votes
2 answers
146 views

$\lim_{s \to 1^+} 1/\zeta(s) = 0$ 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 ...
Vincent's user avatar
  • 10.2k
4 votes
1 answer
134 views

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 ...
nathan benjamin's user avatar
3 votes
0 answers
54 views

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(\...
vitalmath's user avatar
  • 219
0 votes
0 answers
97 views

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{=\...
MathBS's user avatar
  • 3,053
3 votes
1 answer
78 views

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\...
Kishalay Sarkar's user avatar
2 votes
0 answers
62 views

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 ...
Anatoly's user avatar
  • 17k
1 vote
1 answer
106 views

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 ...
Alex Mathy's user avatar
0 votes
1 answer
354 views

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}...
VLC's user avatar
  • 2,519
0 votes
1 answer
1k views

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 ...
Sayan Dutta's user avatar
  • 8,122
1 vote
0 answers
50 views

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 ...
Obsessive Integer's user avatar
1 vote
2 answers
192 views

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 $n=p_1 p_2 ... p_r$, ...
Eug's user avatar
  • 21
2 votes
1 answer
156 views

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^...
Pedja's user avatar
  • 12.8k
1 vote
1 answer
60 views

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 ...
Pedja's user avatar
  • 12.8k
1 vote
0 answers
48 views

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) := \...
dld's user avatar
  • 11
1 vote
1 answer
81 views

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 ...
user avatar
3 votes
0 answers
161 views

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)=...
user avatar
2 votes
1 answer
97 views

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 $\zeta(s)$ ...
Steven Clark's user avatar
  • 6,766

1
2 3 4 5
8