For questions related to Moebius inversion and its applications.

learn more… | top users | synonyms

0
votes
1answer
33 views

Why is this Moebius equivalence true?

I would like to know why the following is true: $$\tau(n^2) = \sum_{d | n} \mu(n/d)(\tau(d))^2$$ I cannot derive it. It is on OEIS but I'd like to know how this was found. $\tau(n)$ is the count of ...
0
votes
1answer
45 views

Stupid Mobius transform problem

I feel that this is a very stupid question to be asking, but I can't figure it out. I've been trying to figure out the Mobius inversion formula, with pretty much no experience in this direction at ...
1
vote
1answer
40 views

moebius transforms preserve sum of signed curvatures

Let $P$ be a point where three arcs of circle meet at equal angles (120 degrees). Suppose that the sum of the curvatures (with sign given by orientation) of the three arcs is zero. Is it true that ...
0
votes
0answers
20 views

Moebius Inversion Question

Let $g$ be an arithmetic function satisfying $\sum_{d|n} g(d) = \frac{n-1}{n+1}$ Find $g(24)$ I have a feeling moebius inversion is going to be used. Really I am just not sure what this is ...
1
vote
1answer
173 views

Number of triplets for which $\gcd(a,b,c)=1$ and $c=n$

As stated in A123323: Number of triples a,b,c with a<=b<=c<a+b, gcd(a,b,c)=1 and c=n. ...... A123323(n)=sumdiv(n, d, floor((d+1)^2/4)*moebius(n/d)) How ...
1
vote
0answers
125 views

Infinite sum of mobius function

Show that if $|q| < 1$, then $\displaystyle{\sum_{n=1}^{\infty}} \frac{\mu(n)*q^n}{1-q^n} = q$. I have a feeling that $\displaystyle{\sum_{n=0}^{\infty}}q^n = \frac{1}{1-q}$ (for $|q|<1$) is ...
2
votes
1answer
136 views

Simple Divisor Sum Transformation by Changing the Order of Double Summation

Show that $$\sum_{d|n} \frac{n}{d} \sigma(d) = \sum_{d|n} d \tau(d)$$ by changing the order of summations from each side to the other. $\sigma$ and $\tau$ are divisor sum functions. ...
6
votes
2answers
146 views

Simple Divisor Summation Inequality (with Moebius function)

Show that $$\left| \sum_{k=1}^{n} \frac {\mu(k)}{k} \right| \le 1 $$ where $\mu$ is Moebius function and n is a positive integer. The hard thing here is that the sum is not directly ...
4
votes
1answer
85 views

Continuous version of the Möbius inversion theorem

Is there a continuous version of Möbius Inversion. Essentially, using integrals instead of sums.
4
votes
1answer
262 views

Rigorously Defining Log of Polynomials?

Below is a proof that the cyclotomic polynomial $\Phi_n(x)=\prod_{d|n}(x^d-1)^{\mu(n/d)}$ using Möbius inversion. However, it requires that we take the log of a polynomial, which (to my knowledge) is ...
0
votes
4answers
289 views

Seeking for a proof on the relation between Euler totient and Möbius function

Can someone help me prove the relation $\varphi\left(n\right)={\displaystyle \sum_{d|n}}d\mu\left(n/d\right)$, where $\mu$ is the Möbius function defined by $$ \mu\left(n\right)=\begin{cases} 1 & ...
0
votes
0answers
48 views

Meaningful examples for $P\left( x\right) $ and $Q\left( x\right) $

Meaningful examples for $P\left( x\right) $ and $Q\left( x\right) $ satisfying $P\left( x\right) =\sum_{n=0}^{\infty }a_{n}\frac{d^{n}}{dx^{n}}% Q\left( x\right) $. I want to find an example for the ...
0
votes
1answer
49 views

Looking for suggestions on how to proceed with showing that:

for $x \ge 2863:$ $$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$ I've written a java application which ...
3
votes
1answer
63 views

Can the Möbius inversion formula be applied to the second Chebyshev function?

Is this a valid application of the Möbius Inversion Formula: Define: $$\psi\left(x\right) = \sum\limits_{p^k \le x} \log p$$ So that: $$\log x! = ...
8
votes
1answer
164 views

Looking for help understanding the Möbius Inversion Formula

I was recently told that the Möbius Inversion Formula can be applied to the Chebyshev Function. Let $\vartheta(x)$,$\psi(x)$ be the first and second Chebyshev functions so that: $$\vartheta(x) = ...
5
votes
1answer
698 views

Monic Irreducible Polynomials over Finite Field

Let $F=\mathbb{F}_{q}$ be a finite field (so $q=p^k$ for some prime $p$ and positive integer $k$), and let $\varphi(d)$ denote the number of monic irreducible polynomials of degree $d$ in $F[X]$. I'm ...
2
votes
2answers
328 views

Show that $\sigma(n) = \sum_{d|n} \phi(n) d(\frac{n}{d})$

This is a homework question and I am to show that $$\sigma(n) = \sum_{d|n} \phi(n) d\left(\frac{n}{d}\right)$$ where $\sigma(n) = \sum_{d|n}d$, $d(n) = \sum_{d|n} 1 $ and $\phi$ is the Euler Phi ...
7
votes
0answers
161 views

Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
2
votes
1answer
250 views

Existence of an irreducible polynomial over $\mathbb F_p$. [duplicate]

Possible Duplicate: Lack of understanding of the proof of the existence of an irreducible polynomial of any degree $n \geq 2$ in $\mathbb{Z}_p[x]$ Existence of irreducible polynomials over ...
3
votes
2answers
652 views

Sum of primitive roots is congruent to $\mu(p-1)$ using Moebius inversion?

Wikipedia has the result that Gauss proved that for a prime number $p$ the sum of its primitive roots is congruent to $\mu(p − 1) \pmod{p}$ in Article 81. I read it, but is there a faster proof ...
2
votes
1answer
181 views

Question on Euler Totient, and finding a function

Determine $f(n)$ such that for all $n\geq 1$, $$\frac{1}{\varphi (n)}=\sum_{d\vert n}\left(\frac{1}{d}\right)f\left(\frac{n}{d}\right)$$ This is not a homework question, just a question I stumbled ...
1
vote
2answers
935 views

Are all finite groups cyclic?

I've been reading about Number Theory and I came across this proof that the finite subgroups of the multiplicative group of a field is cyclic. However, it seems the proof applies to all finite groups ...