Questions tagged [mobius-function]

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

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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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  • 337
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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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2 votes
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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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what is the mobius transformation that maps a circle with center $z_0$ and radius $R$ to the unit circle? [closed]

I want to find the mobius function $f(z)$ that transforms the circle with center $z_0$ and radius $R$ to the unit circle centered at the origin.
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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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Calculate Mertens Function in Sublinear Time

I'd like to calculate the Mertens function $M(n)=\sum_{i=1}^{n} \mu(i)$ in sublinear time using the following formula: $M(n)=M(\lfloor \sqrt{n} \rfloor)-\sum_{1 \leq i,j \leq \lfloor \sqrt{n} \rfloor} ...
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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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Is the linear property of the sequence that contains sums of Möbius function values explainable/provable?

Let $\mu(n)$ be the Möbius function. We denote $M(x)=\sum_{n=1}^x\mu(n)$ as the sum of Möbius function values from $n=1$ up to $x$. Mikolás proved in his artice Farey series and their connection with ...
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1 vote
2 answers
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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 ...
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If n is even, prove that the summation (indexed over the divisors of n) ϕ(d)µ(d) = 0 [duplicate]

I am having great difficulty with the following proof: Prove that if $n$ is even, $\sum_{d|n} μ(d)ϕ(d) = 0$ First, I noticed a general pattern that we will use later: For any integer $a$, we see that ...
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3 votes
0 answers
157 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)=...
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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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Is this relation already discovered?

$$ \sum_{d \mid (n,k_1,k_2, \dots,k_m)}\mu(d)\binom{n/d}{k_1/d, k_2/d, \dots, k_m/d} \equiv 0 \pmod n $$ where $\mu$ is the Moebius mu function. I've found above interesting divisibility properties. I'...
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Show that, $(-1)^{\mu(1)}+(-1)^{\mu(2)}+...+(-1)^{\mu(n)}<0$ and proof on conjecture of OEIS A209802

Following is an experimental math claim. We denote $\mu(a)$ as Möbius function Let $$F(a)=\sum_{i=1}^{a}(-1)^{\mu(i)}.$$ Can it be shown that for every positive integer $a$, $F(a)<0$? Table $$\...
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Do these multiplicative functions exist?

I read about the multiplicative function $\mu$ defined by Möbius. Which is for any given $n\in \mathbb{N}$ such as $n=p_1^{\alpha_1}\cdot ....\cdot p_r^{\alpha_r}$ is defined as $$\mu(n)= \left\{ \...
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1 answer
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Möbius function of a linearly (totally) ordered set

Let $I_n$ be a totally ordered set ${1<2<...<n}$. We need to calculate $\mu(m,n)$. We shall use the recursive definition of $\mu$. We have the following: $$\mu(m,m) = 1$$ $$\mu(m,m+1) = -\...
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Dirichlet convolution of Mobius function with exponential function

Define $\exp_x : \mathbb{N} \rightarrow \mathbb{C}$ by $\exp_x(d) = e^{ixd}$ for all $d \in \mathbb{N}$ and some $x \in \mathbb{R}$. I want to evaluate the Dirichlet convolution of the Mobius function ...
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Showing $\mu(n) = \sum_{k} e^{2\pi \frac{k}{n} i}$ using cosine explanation [duplicate]

Who closed this question? the similar one is actually different, if you see the answer. The mobius function $ \mu (n)$, is a function with property: $$\mu(1) = 1$$ $$\mu(n) = 0$$ if $n$ is of the ...
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Sums of Möbius between $x$ and $y$

For $z_1 > z_2 \geq 0$ define $$M(z_1,z_2) = \sum_{z_2 < a \leq z_1 } \mu(a),$$ where $\mu$ is the Möbius' function. Prove that $$\sum_{k=1}^{\infty} M\left(\frac{n}{k}, 0\right) = 1\,\text{ and ...
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1 vote
1 answer
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Solutions to the following equation including a Mobius function

Let $\mu$ be the Mobius function. Now, how would one solve the following equation with respect to the variables $a\in \mathbb{N}$ and $b\in \mathbb{N}$? $\sum_{d=1}^{\infty}\mu(d)\lfloor \frac{a}{d}\...
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5 votes
2 answers
121 views

Is there an unlimited number of sequences of consecutive numbers with $\mu(n)=0$ of any length?

EDIT I have received interesting comments to my post. Especially the comment of @Martin Hopf showed that this problem is a "classic" and by no means new. Here is Eric Weissten's article on &...
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1 answer
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Convolving the relation $o(x)$

I'll get right to the point. For sequences $\{a_n\}$ and $\{b_n\}$ in $\mathbf C$, let $c_n = \sum_{d \mid n} a_db_{n/d}$. Set $A(x) = \sum_{n \leq x} a_n$, $B(x) = \sum_{n \leq x} b_n$, and $C(x) = \...
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8 votes
2 answers
139 views

Squarefree totient sum

Does anybody have a reference/proof for the asymptotic growth rate of $$A(x) = \!\!\!\!\!\!\sum_{\substack{n \leqslant x \\ n \ \text{squarefree}}} \!\!\!\!\!\! \varphi(n)$$ as $x \to \infty$? Here $\...
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0 votes
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Möbius function for prime $p$ and gcd of prime $p$ and $d$ where $d$ divides $n$ [closed]

Let $\mu(p,d)$ denote the value of the Möbius function at the gcd of $p$ and $d$. Prove that for every prime $p$ we have $$\sum_{d|n}\mu(d)\mu(p,d) = \begin{cases} 1 & \text{if $n=1,$} \\ 2 &...
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2 votes
1 answer
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Number Theory: Möbius Function

Background Definition: If $S$ with partial order ≤ is locally finite with minimal element, then the generalized Möbius function $\mu$: S $\times$ S $\rightarrow$ $\mathbb{N}$ by (a) $\forall$ s $\in$...
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Under which circumstance is $\left| \sum_{n = 1}^{x}\mu(n)f(n) \right| \ge \left| \sum_{n = 1}^{x}\mu(n)g(n) \right| $

I would appreciate assistance in either verifying this conjecture, or perhaps a source relating to similar inequalities involving the Mobius function. $\left| \sum_{n = 1}^{x}\mu(n)f(n) \right| \ge \...
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1 vote
0 answers
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Is this sequence in one to one correspondence with the Mertens function?

Consider the sequence $s(n)=$ $$\small \frac{\left(\prod _{b=2}^{\frac{n}{2^1}} \prod _{a=2}^{\frac{n}{2^1}} (1+i [a b\leq n])\right) \left(\prod _{d=2}^{\frac{n}{2^3}} \prod _{c=2}^{\frac{n}{2^3}} \...
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2 votes
0 answers
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Can anything useful be said about or done with an obvious generalisation of the Mobius function?

For a natural number $n$, the standard Mobius function $\mu(n)$ is defined to be zero if the square of any prime divides $n$, and otherwise either $-1$ or $1$ according as n has respectively an odd or ...
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1 answer
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Understanding the values of Möbius function

I want to understand how did he get the values for this intersection poset: So I know that he used the Möbius function but I don't understand how he get the values only using the following : But how ...
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0 votes
1 answer
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Convergence of the double series $\sum_{d|n} \mu(d) x_{n}$

Let $\sum x_n$ be a power series which behaves sufficiently nicely, for example, absolutely convergent. Can we deduce that the double series $$ \sum_{d|n} \mu(d) x_{n} $$ converges in the sense ...
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0 votes
1 answer
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Dirichlet convolution of the small prime omega function and the Mobius function

I have seen that: $$(\omega\star\mu)(n)=\sum_{d\vert n}\mu(d)\omega\left(\frac{n}{d}\right)=\begin{cases}1 & n\ \text{is prime}\\ 0 &\text{otherwise} \end{cases}$$ where $\mu(n)=\delta_{\omega(...
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3 votes
1 answer
78 views

Möbius function of subpermutation ordering of $[n]$, is equal to number of derangements of $[n]$.

Let $P_n$ be the poset of subpermutations of $[n]$, ordered by the relation $$x\preceq y \iff x \text{ is a subsequence of } y$$ (e.g. for $P_7$, we have that $(3,7,6)\preceq (2,3,7,1,6)$ ). Let us ...
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0 votes
1 answer
142 views

Prove that $\sum_{n=1}^{\infty} \frac{\mu(n)}{10^n}$ is irrational

First of all, I'm aware that this question has been previously asked, (see: show that $\sum \frac {\mu(n)}{10^n}$ is irrational) however I did not find the solutions there particularly useful. In ...
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2 votes
2 answers
119 views

What does $\sum \mu(d)\lfloor{\frac{x}{d}}\rfloor$ count?

Where the sum is over all $d$, where $d$ only has prime factors $\leq \sqrt{x}$. I'm trying to determine, in plain English, precisely what this sum is counting. I know the Möbius function will be 0 if ...
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0 votes
1 answer
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Möbius function - average

I had to solve an exercise comprised of 3 parts: a) For q, a integers, $q\geq 2$ and $\gcd(a,q)=1$ to show that $$\lim_{x \rightarrow \infty} \frac{1}{x}\mkern-18mu \sum_{\substack{n \leq x\\n \equiv ...
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0 votes
1 answer
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Help proving that $\sum_{d^k|n}\mu(d) =$ 1 if is k-free and 0 otherwise

In class we saw a similar result when $k=2$, and now I'm trying to extend this to an arbitrary $k\in \mathbb{Z}^+$. When I plug in values this identity seems to hold, however I'm unsure how to tackle ...
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1 vote
0 answers
42 views

Any results about auto-correlation function of Mertens function

One can define $f(x) = M(e^x)/\sqrt{e^x}$, where $M$ is Mertens function. It looks like some sort of stationary random process (yes, I know it's not random process, see plot below), namely it lives ...
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2 votes
1 answer
144 views

A matrix related to the möbius function

Consider the matrix $A_n$ defined for positive integers $n$ by setting the $(i,j)$th entry to $1$ if $j$ divides $i$, and $0$ otherwise, for $1\leq i,j\leq n$. For example, $$A_6=\begin{bmatrix}1&...
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2 votes
1 answer
159 views

Counting Squarefree Integers $i \le n$ Coprime to the First $k$ Primes

The number of positive squarefree integers $i \le n$ is given by: $$C(n)=\sum_{k=1}^{\lfloor\sqrt{n}\rfloor}\mu(k)\left\lfloor\frac{n}{k^{2}}\right\rfloor.$$ The number of positive integers $i\le n$ ...
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