# Questions tagged [mobius-function]

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

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### Summation of certain divisible numbers

Let $C(N,m)$ be the number of positive integers $\le N$ which are relatively prime to $m$. It can be found by following equation \begin{align} C(N,m)=\sum_{d|m}\mu (d)\left\lfloor\frac{N}{d}\right\...
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### Tight bounds on the partial Möbius sum $\sum_{\substack{d|n\\d<Q}}\mu(d)$

An important area of study in Analytic Number Theory is the behavior of the Möbius function $\mu(n)$. I was trying to prove a different theorem when I came about a very interesting behavior. If you ...
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### Mobius Function

Find all integers $n, 1 ≤ n ≤ 100$ such that $µ(n) = 1$. My initial idea was to find all the primes between 1 and 100 using techniques of finding prime numbers. If p is prime then $µ(p)=-1$ so with ...
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### Is anything known about bounds of $\sum_{n<x} \frac{\mu(n)}{\sqrt{n}}$?

Without assuming Riemann hypothesis, is anything known about bounds of partial sum $\sum_{n<x} \frac{\mu(n)}{\sqrt{n}}$ where $\mu(n)$ is Möbius function and $n \in \mathbb{N}$. It is known that ...
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### Mobius inversion on the partition lattice

For some $n \in \mathbb N$, let $(\Pi_n, \le)$ be the poset of partitions of the set $\{1, 2, \dots, n\}$, where two partitions $\pi, \rho \in \Pi_n$ have the relation $\pi \le \rho$ if $\pi$ is a ...
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### Determine the number of irreducible polynomials of degrees 2, 3, and 6 over a prime ﬁeld.

Determine the number of irreducible polynomials of degrees 2, 3, and 6 over the prime ﬁeld $\mathbb F_p$. My question is related to the answer posted for the above question. I was scratching my head ...
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### Can we prove that partial sum $|\sum_1^N \mu(n)/n| \le 1/2$ when $N > 1$?

Here, $\mu(n)$ is Möbius function. From this, we can see that partial sum $|\sum_1^N \frac{\mu(n)}{n}| \le 1$. when $N \in \mathbb{N}$. We also know the series converges to $0$ and is not monotone. ...
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### Möbius function of partition lattice

I know we can use Weisner's Theorem to obtain $\mu(\Pi_n)=(-1)^{n-1}(n-1)!$ where $\Pi_n$ is the partition lattice on an $n$-set. However, I was wondering if we could derive this result using the ...
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### How to calculate the GCD?

How to evaluate the following with the help of Mobius function ? $$\displaystyle\sum_{i=1}^n \sum_{j=i+1}^n \sum_{k=j+1}^n \sum_{l=k+1}^n {gcd(i,j,k,l)^4} .$$ In other words, we have to ...
I've verified conjectured relationship (3) below for several thousand values of $x$ and suspect that it's true. (1) $\quad H(x)=\sum\limits_{n=1}^x\frac{1}{n}\qquad\qquad\qquad\quad$(Harmonic Number ...
Define the collection of sets $\left\{Q_n\right\}$ as follows: $$Q_1 = \{0\};\quad Q_2 = \left\{\frac{1}{2}\right\} \cup Q_1; \quad Q_3 = \left\{\frac{1}{3}, \frac{2}{3}\right\} \cup Q_2;$$ \left\{\...