# 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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### 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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### 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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### 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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### 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 ...
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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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### 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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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 $$\sum_{k=1}^{\infty} M\left(\frac{n}{k}, 0\right) = 1\,\text{ and ... • 2,537 1 vote 1 answer 32 views ### Solutions to the following equation including a Mobius function Let \mu be the Mobius function. Now, how would one solve the following equation with respect to the variables a\in \mathbb{N} and b\in \mathbb{N}? \sum_{d=1}^{\infty}\mu(d)\lfloor \frac{a}{d}\... 5 votes 2 answers 121 views ### Is there an unlimited number of sequences of consecutive numbers with \mu(n)=0 of any length? EDIT I have received interesting comments to my post. Especially the comment of @Martin Hopf showed that this problem is a "classic" and by no means new. Here is Eric Weissten's article on &... • 10.3k 3 votes 1 answer 78 views ### Convolving the relation o(x) I'll get right to the point. For sequences \{a_n\} and \{b_n\} in \mathbf C, let c_n = \sum_{d \mid n} a_db_{n/d}. Set A(x) = \sum_{n \leq x} a_n, B(x) = \sum_{n \leq x} b_n, and C(x) = \... • 30.1k 8 votes 2 answers 139 views ### Squarefree totient sum Does anybody have a reference/proof for the asymptotic growth rate of$$A(x) = \!\!\!\!\!\!\sum_{\substack{n \leqslant x \\ n \ \text{squarefree}}} \!\!\!\!\!\! \varphi(n)$$as x \to \infty? Here \... • 7,401 0 votes 1 answer 95 views ### Möbius function for prime p and gcd of prime p and d where d divides n [closed] Let \mu(p,d) denote the value of the Möbius function at the gcd of p and d. Prove that for every prime p we have$$\sum_{d|n}\mu(d)\mu(p,d) = \begin{cases} 1 & \text{if $n=1,$} \\ 2 &...
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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 $\in$...