Questions tagged [mobius-inversion]
The mobius-inversion tag has no usage guidance.
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Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
$$\Lambda(n)=...
27
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0
answers
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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 ...
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Calculating Mayer-Vietoris efficiently
This is a question whose motivation and framing seem to involve a lot of topology, but which I suspect comes down to some simple and standard combinatorics that's probably recorded in a book somewhere....
- 151k
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2
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Equivariant Möbius inversion
I'll first explain what Möbius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is ...
- 151k
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0
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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 ...
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2
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Alternating sum over collections closed under containment
Let $\mathscr{C}$ be a collection of subsets of a finite set $P$. Assume $\mathscr{C}$ is closed under containment: if $S\subset P$ is in $\mathscr{C}$, then every set $S'\subset P$ containing $S$ is ...
- 19.4k
8
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What is the Möbius function for the lattice of partial partitions?
Let $n$ be a positive integer. Let $P$ be the set of partitions of subsets of
$\{ 1, 2, \dotsc, n \}$ (so, for example, when $n=2$, the set $P$ contains $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{...
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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 }...
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4
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How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$?
I'm actually interested in a slightly smaller quantity, but I'm willing to accept the following simplification, especially if there are small error terms.
Let's start with $ n \gt 1$, Euler's totient ...
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What is the Möbius function of substrings?
Define a poset on the set of all finite binary strings, defined by $a \le b$ whenever $b = uav$ for (possibly empty) binary strings $u, v$.
What is the Möbius function of this poset?
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How often does the Mertens function vanish?
It is well known that the Mertens function
$$M(x)=\sum _{n\leq x}\mu(n)$$
has infinitely many zeros, and this seems to be a short proof.
Are there known results about how often the Mertens function ...
- 577
6
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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 ...
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What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...
- 1,133
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2
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Problems with the divisor function in a summation
I'm trying to work with the following sum:
$$f:=\sum_{d\leq x}\mu(d)\tau(d) \Big[ \frac{x}{d}\Big] $$
Where $\mu$ is the Mobius function, $\tau(n)$ is the number of positive divisors of $n$ and $h(x)=[...
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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 $...
- 1,121
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1
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Seeking combinatorial interpretation of a quantity that comes up from central hyperplane arrangements
Let $\mathcal A$ be a central hyperplane arrangement in $\mathbb R^d$ and let's assume that it is essential, meaning the hyperplanes in $\mathcal A$ intersect in the origin. The intersection lattice $...
- 37.4k
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3
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Alternating sum over collections of sets
Let $\mathbf{P}$ be a collection of subsets of a finite set $X$. Let $\mathscr{S}$ be the set of all subsets $\mathbf{S}\subset \mathbf{P}$ such that $\bigcup_{S\in \mathbf{S}} S = X$. Can one give a ...
- 19.4k
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Möbius inversion formula and roots of unity
Is the exact value of
$$
\sum_{d\mid n} \mu\left(\frac{n}{d}\right) \zeta^d
$$
known? Here, $\mu$ denotes the Möbius function and $\zeta$ a root of unity. At first sight, it seems to me that this ...
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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 ...
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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 $...
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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
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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
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0
answers
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A chain of six circles associated with six points on a circle (in Mobius plane) [closed]
I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane).
This is a generalization of the last my previous question in Three chains of six circles. (Noting ...
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1
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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}^\...
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Largest value of the Möbius function for subposet of product of chains
Given a product $P$ of chains of lengths $a_1, \dots, a_n$, what is an upper bound on the largest possible value of the absolute value of the Möbius function on a subposet of this poset? Perhaps in ...
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Trace formula for the Möbius function
I found this conjecture while working with the Möbius function
$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{...
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Finding a Big Theta Bound for a Summation Involving the Möbius Inversion Formula
I am currently in the midst of a summer research project and have run across an interesting summation: $F(n) = \sum\limits_{i=1}^{\lfloor\frac{n}{2}\rfloor}(n - 2i + 1)P_2(i)$.
And here are some ...
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Asymptotic behaviour of a sum involving Möbius function
(This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.)
I am trying to get the asymptotic behaviour when $n$ grows to infinity of a partial sum of ...
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Integers with $k$ prime factors, in terms of the Möbius function
A $k$-free integer is an integer $n$ such that there is no $k$th power dividing $n$. It is well known (see Murty's Problems in Analytic Number Theory q1.18 for instance) that \begin{equation}\sum_{d^k|...
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Mobius function for reflection subgroups
Let $W$ be a finite real reflection group. Let $\Pi(W)$ be the set of reflection subgroups of $W$. Then $\Pi(W)$ has a natural partial order defined by inclusion. Let $\mu$ denote the associated ...
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Prove that: $\sum _{c=1}^n \sum _{b=1}^n \sum _{a=1}^n \left(\left([b|c][b|a]\frac{\mu(b)b}{a}\right)-\frac{1}{a b\sqrt{c}}\right)<H_n+n$
In the OEIS there is the quote from Lowell Schoenfeld that the Riemann hypothesis is equivalent to:
$$|\psi(n) - n| < \sqrt{n} \log^2(n)$$
From the Euler Maclaurin formula one gets:
$$\sum _{c=1}^n ...
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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. ...
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can this sum be true ??
$$ \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}} g \log n = \sum_t \frac{h(t)}{\zeta'(1/2+it)}+2\sum_{n=1}^\infty \frac{ (-1)^{n} (2\pi )^{2n}}{(2n)! \zeta(2n+1)}\int_{-\infty}^{\infty}g(x) e^{-x(2n+1/2)}...
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0
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Minimum diameter of spherically-inverted topological balls
Let $U$ be the closed unit ball in $\mathbb{R}^3$. Let $S$ be a round sphere whose center is in $U$ with radius at least $\delta_1 > 0$. Suppose $B$ is a closed topological ball of Euclidean ...
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A question about the Heilbronn-Rohrbach Inequality
Let $\Phi(x,p)$ = the number of integers $i$ where $0 < i \le x$ and $\gcd(x,p)=1$.
Let $p\#$ be the primorial for $p$.
Using the inclusion-exclusion principle:
$$\Phi(x,p\#) = \sum_{d|p\#}\mu(d)...
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