Questions tagged [mobius-inversion]
The mobius-inversion tag has no usage guidance.
18
questions with no upvoted or accepted answers
25
votes
0
answers
540
views
A conjecture about inclusion–exclusion
$\newcommand\calF{\mathcal{F}}
\def\cupdot {\stackrel{\bullet}{\cup}}
\def\minusdot {\stackrel{\bullet}{\setminus}}$This post presents a conjecture that we have with some colleagues. It is about ...
13
votes
0
answers
1k
views
A question about Mobius inversion
I don't know how precise I can make this question. I want to know whether there is a theorem that says that a certain phenomenon always happens, but I think the best I can do in order to pin down the ...
7
votes
0
answers
889
views
The Möbius function as eigenvalues
Let the $N$ by $N$ matrix $A$ be defined by the tetration:
$$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
6
votes
0
answers
222
views
Gaussian coefficients identity
I am having difficulty showing the equivalence between (11) and (15) of Delsarte - Association schemes and $t$-designs in regular semilattices. It is somehow an application of Möbius inversion, but I ...
3
votes
0
answers
432
views
Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?
My question is on Mobius inversion formula convergence/properties when used with infinite sums of function.
Lets consider (on $\mathbb{R}^{+}$):
$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$
We call $...
2
votes
0
answers
122
views
Proof of Crapo's complementation theorem
In Crapo's work "Möbius inversion in lattices," he gave a second proof of his complementation theorem:
$$ \mu(0,1) = \sum_{x, y \in s^\perp} \mu(0,x) \zeta(x, y) \mu(y,1)$$
where $s$ is an ...
2
votes
0
answers
186
views
On a generalization of the Möbius function from number theory
Let $\omega$ be a positive real number, and define: $$\mathbf{1}_{\omega}\left(n\right)\overset{\textrm{def}}{=}\left(-1\right)^{n}\binom{-\omega}{n}=\binom{\omega-1+n}{n}$$ for all positive integers $...
2
votes
0
answers
82
views
A lattice ordered by inclusion and isomorphic to the lattice of quotient groups of a finite group
Let $G$ be a finite group. Consider the lattice $$L=\{ G/N:\text{$ N $ is a normal subgroup of $G $}\},$$ where $G/N \leq G/K$ if and only if $K\leq N$. The lattice operations ∧ and ∨ on quotient ...
2
votes
0
answers
156
views
On the divisor function in a summation
I want to compute the limit inferior and superior of the following sum
$$f(x):=\sum_{\substack{d \leq x\\P^+(d)\le\sqrt{x}}} \mu(d)\tau(d)\left[\frac{x}{d}\right]$$
Where $μ$ is the Mobius function, $...
2
votes
1
answer
530
views
On infinite sum containing logarithmic derivative of Zeta function and Möbius function:
Consider the following function:
$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$
Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...
1
vote
0
answers
182
views
Asymptotic behaviour of a sum involving Möbius function
(This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.)
I am trying to get the asymptotic behaviour when $n$ grows to infinity of a partial sum of ...
1
vote
0
answers
158
views
Integers with $k$ prime factors, in terms of the Möbius function
A $k$-free integer is an integer $n$ such that there is no $k$th power dividing $n$. It is well known (see Murty's Problems in Analytic Number Theory q1.18 for instance) that \begin{equation}\sum_{d^k|...
1
vote
0
answers
89
views
Mobius function for reflection subgroups
Let $W$ be a finite real reflection group. Let $\Pi(W)$ be the set of reflection subgroups of $W$. Then $\Pi(W)$ has a natural partial order defined by inclusion. Let $\mu$ denote the associated ...
1
vote
0
answers
149
views
Prove that: $\sum _{c=1}^n \sum _{b=1}^n \sum _{a=1}^n \left(\left([b|c][b|a]\frac{\mu(b)b}{a}\right)-\frac{1}{a b\sqrt{c}}\right)<H_n+n$
In the OEIS there is the quote from Lowell Schoenfeld that the Riemann hypothesis is equivalent to:
$$|\psi(n) - n| < \sqrt{n} \log^2(n)$$
From the Euler Maclaurin formula one gets:
$$\sum _{c=1}^n ...
1
vote
0
answers
112
views
Zero mean constraint for correlations with the Mobius function
An aspect of my research has lead me to believe that I need to distinguish between those bounded functions $\xi:\mathbb{N}\rightarrow\mathbb{C}$ which correlate with the Mobius function $\mu(n)$, i.e. ...
1
vote
0
answers
180
views
can this sum be true ??
$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)}...
0
votes
0
answers
33
views
Minimum diameter of spherically-inverted topological balls
Let $U$ be the closed unit ball in $\mathbb{R}^3$. Let $S$ be a round sphere whose center is in $U$ with radius at least $\delta_1 > 0$. Suppose $B$ is a closed topological ball of Euclidean ...
0
votes
0
answers
141
views
A question about the Heilbronn-Rohrbach Inequality
Let $\Phi(x,p)$ = the number of integers $i$ where $0 < i \le x$ and $\gcd(x,p)=1$.
Let $p\#$ be the primorial for $p$.
Using the inclusion-exclusion principle:
$$\Phi(x,p\#) = \sum_{d|p\#}\mu(d)...