# 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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### 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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### 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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### 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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### 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)... 52 views

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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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### 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 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 ... ### 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)=... ### 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 $\zeta(s)$ ...