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# Questions tagged [mobius-function]

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

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### Bounding $S(n) = \sum_{k=1}^n \mu(k) \left( \pi\left(\frac{n}{k}\right) - \pi(\text{gpf}(k)) \right)$

I am trying to bound the sum $$S(n) = \sum_{k=1}^n \mu(k) \left( \pi\left(\frac{n}{k}\right) - \pi(\text{gpf}(k)) \right)$$ In other site, I have been given the following "proof". I would ...
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### Using the binomial formula in the form $(k - k)^n$

While proving a certain property of the number theoretic mobius function, namely that it is invertible in the monoid of multiplicative functions and its inverse is ...
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### lower bound of $\sum_{n=1}^x \frac{\mu(n)}{n}$

Denote by $\mu$ the Mobius function. Poussin showed that $$\sum_{n=1}^x \frac{\mu(n)}{n} = O(1/\log x),$$ and there are further improvements since. I wonder what is known about lower bound of ...
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### Why $\prod\limits_{n=1}^{\infty} \biggl (\phi(q^n)^{\mu(n)} \biggr)= 1-q$?

Playing with Euler $\phi$ function (not to be confused with the totient function, here another reference), I found this curious identity (I calculated it for various $q$ with Mathematica and it holds)...
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### Comparing two series expressions for $1/\zeta(s)$. What can be said about their complex roots?

The following two expressions involving the inverted Riemann $\zeta(s)$ functions are well known: \begin{align} \frac{1}{\zeta(s)} &= \sum_{n=1}^\infty \frac{\mu(n)}{n^s} \\ -\frac{\zeta'(s)}{\...
• 3,201
2 votes
1 answer
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### Möbius function of distributive lattice only takes values $\pm 1$ and $0$.

In this Wikipedia article, I found the statement [...] shares some properties with distributive lattices: for example, its Möbius function takes on only values 0, 1, −1. My question is: How it can ...
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### Riemann Hypothesis follows from the statement $M\left(x\right)=o_x(x^{\frac{1}{2}+\varepsilon})$

Recall that the Mertens function is defined via: $$M(n):=\sum_{n\ge x\ge 1} \mu(x)$$ Where $\mu$ is the Möbius function. Littlewood proved that if $M\left(x\right)=o_x(x^{\frac{1}{2}+\varepsilon})$ ...
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### Meaning of $M(n)=O\left(x^{\frac{1}{2}+\epsilon}\right)$

I am trying to fully understand the implications of $M(n)=O\left(n^{\frac{1}{2}+\epsilon}\right)$, where $M(n)$ is Mertens function, being equivalent to Riemann Hypothesis. (i) Is the equivalence ...
• 1,190
2 votes
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### Möbius inversion formula and $\sum_{k\leq n} \frac{\mu(k)}{k}$

I have tried to apply what is stated at the Generalizations of Möbius inversion formula section of Wikipedia to bound $$\sum_{k\leq n} \frac{\mu(k)}{k}$$ The application seems simple and ...
• 1,190
4 votes
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### Bounding a partial sum with Möbius inversion formula

I am trying to bound the partial sum $$S(n)=\sum_{k=1}^{\sqrt{n}} \mu(k) \pi\left(\frac{n}{k}\right)$$ Where $\pi(x)$ is the prime counting function, and $\mu(x)$ is the Möbius function. Empirical ...
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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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### 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 ...
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### 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 ...
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2 votes
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1 vote
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• 199
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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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4 votes
1 answer
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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 ...
3 votes
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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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