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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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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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.
user830472
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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 ...
user736865
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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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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
$$\mu(n)= \left\{ \...
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Möbius function of a linearly (totally) ordered set
Let $I_n$ be a totally ordered set ${1<2<...<n}$. We need to calculate $\mu(m,n)$.
We shall use the recursive definition of $\mu$.
We have the following:
$$\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
$$\sum_{k=1}^{\infty} M\left(\frac{n}{k}, 0\right) = 1\,\text{ and ...
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Solutions to the following equation including a Mobius function
Let $\mu$ be the Mobius function. Now, how would one solve the following equation with respect to the variables $a\in \mathbb{N}$ and $b\in \mathbb{N}$?
$\sum_{d=1}^{\infty}\mu(d)\lfloor \frac{a}{d}\...
user830143
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Is there an unlimited number of sequences of consecutive numbers with $\mu(n)=0$ of any length?
EDIT
I have received interesting comments to my post. Especially the comment of @Martin Hopf showed that this problem is a "classic" and by no means new.
Here is Eric Weissten's article on &...
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Convolving the relation $o(x)$
I'll get right to the point. For sequences $\{a_n\}$ and $\{b_n\}$ in $\mathbf C$, let $c_n = \sum_{d \mid n} a_db_{n/d}$. Set $A(x) = \sum_{n \leq x} a_n$, $B(x) = \sum_{n \leq x} b_n$, and $C(x) = \...
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Squarefree totient sum
Does anybody have a reference/proof for the asymptotic growth rate of
$$A(x) = \!\!\!\!\!\!\sum_{\substack{n \leqslant x \\ n \ \text{squarefree}}} \!\!\!\!\!\! \varphi(n)$$
as $x \to \infty$? Here $\...
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Möbius function for prime $p$ and gcd of prime $p$ and $d$ where $d$ divides $n$ [closed]
Let $\mu(p,d)$ denote the value of the Möbius function at the gcd of
$p$ and $d$. Prove that for every prime $p$ we have
$$\sum_{d|n}\mu(d)\mu(p,d) = \begin{cases} 1 & \text{if $n=1,$} \\ 2
&...
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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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Is this sequence in one to one correspondence with the Mertens function?
Consider the sequence $s(n)=$
$$\small \frac{\left(\prod _{b=2}^{\frac{n}{2^1}} \prod _{a=2}^{\frac{n}{2^1}} (1+i [a b\leq n])\right) \left(\prod _{d=2}^{\frac{n}{2^3}} \prod _{c=2}^{\frac{n}{2^3}} \...
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Can anything useful be said about or done with an obvious generalisation of the Mobius function?
For a natural number $n$, the standard Mobius function $\mu(n)$ is defined to be zero if the square of any prime divides $n$, and otherwise either $-1$ or $1$ according as n has respectively an odd or ...
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Understanding the values of Möbius function
I want to understand how did he get the values for this intersection poset:
So I know that he used the Möbius function but I don't understand how he get the values only using the following :
But how ...
user730480
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Convergence of the double series $\sum_{d|n} \mu(d) x_{n}$
Let $\sum x_n$ be a power series which behaves sufficiently nicely, for example, absolutely convergent. Can we deduce that the double series
$$
\sum_{d|n} \mu(d) x_{n}
$$
converges in the sense ...
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Dirichlet convolution of the small prime omega function and the Mobius function
I have seen that: $$(\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}$$ where $\mu(n)=\delta_{\omega(...
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Möbius function of subpermutation ordering of $[n]$, is equal to number of derangements of $[n]$.
Let $P_n$ be the poset of subpermutations of $[n]$, ordered by the relation $$x\preceq y \iff x \text{ is a subsequence of } y$$ (e.g. for $P_7$, we have that $(3,7,6)\preceq (2,3,7,1,6)$ ).
Let us ...
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Prove that $\sum_{n=1}^{\infty} \frac{\mu(n)}{10^n}$ is irrational
First of all, I'm aware that this question has been previously asked, (see: show that $\sum \frac {\mu(n)}{10^n}$ is irrational) however I did not find the solutions there particularly useful.
In ...
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What does $\sum \mu(d)\lfloor{\frac{x}{d}}\rfloor$ count?
Where the sum is over all $d$, where $d$ only has prime factors $\leq \sqrt{x}$. I'm trying to determine, in plain English, precisely what this sum is counting.
I know the Möbius function will be 0 if ...
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Möbius function - average
I had to solve an exercise comprised of 3 parts:
a) For q, a integers, $q\geq 2$ and $\gcd(a,q)=1$ to show that
$$\lim_{x \rightarrow \infty} \frac{1}{x}\mkern-18mu \sum_{\substack{n \leq x\\n \equiv ...
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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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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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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,
$$A_6=\begin{bmatrix}1&...
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Counting Squarefree Integers $i \le n$ Coprime to the First $k$ Primes
The number of positive squarefree integers $i \le n$ is given by: $$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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