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When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?

This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
Roger Van Peski's user avatar
27 votes
0 answers
610 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 ...
M.Monet's user avatar
  • 371
2 votes
0 answers
206 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 ...
Juan Moreno's user avatar
1 vote
0 answers
165 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|...
alidixon222's user avatar
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 ...
maxematician's user avatar
1 vote
1 answer
170 views

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 ...
johng23's user avatar
  • 270
8 votes
1 answer
303 views

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 $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{...
Naysh's user avatar
  • 507
2 votes
0 answers
125 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 ...
yaoliang's user avatar
  • 101
1 vote
0 answers
93 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 ...
Dr. Evil's user avatar
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6 votes
0 answers
223 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 ...
Leon Bankston's user avatar
2 votes
1 answer
178 views

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 $...
Benjamin Steinberg's user avatar
6 votes
1 answer
424 views

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?
user176357's user avatar
2 votes
1 answer
539 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}^\...
Zaza's user avatar
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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 ...
Mats Granvik's user avatar
  • 1,133
2 votes
0 answers
188 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 $...
MCS's user avatar
  • 1,274
2 votes
3 answers
472 views

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 ...
H A Helfgott's user avatar
  • 19.3k
9 votes
2 answers
403 views

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 ...
H A Helfgott's user avatar
  • 19.3k
2 votes
1 answer
328 views

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 ...
LFM's user avatar
  • 171
5 votes
3 answers
1k views

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 $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
Mats Granvik's user avatar
  • 1,133
7 votes
0 answers
898 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 }...
Mats Granvik's user avatar
  • 1,133
6 votes
2 answers
765 views

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 ...
Basj's user avatar
  • 577
17 votes
2 answers
729 views

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 ...
David E Speyer's user avatar
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 ...
Farid Aliniaeifard's user avatar
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, $...
ADLEN A AYADI's user avatar
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)...
Larry Freeman's user avatar
1 vote
0 answers
113 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. ...
Kevin Smith's user avatar
  • 2,470
2 votes
0 answers
268 views

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 ...
Oai Thanh Đào's user avatar
6 votes
4 answers
990 views

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 ...
Gerhard Paseman's user avatar
3 votes
0 answers
436 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 $...
Bertrand's user avatar
  • 1,121
49 votes
2 answers
10k views

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)=...
1 vote
1 answer
436 views

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 )^{...
user23964's user avatar
  • 113
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)}...
jose's user avatar
  • 11
4 votes
2 answers
1k views

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)=[...
Alex Botros's user avatar
1 vote
1 answer
635 views

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 ...
lekkin's user avatar
  • 13
23 votes
2 answers
3k views

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....
David E Speyer's user avatar
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 ...
gowers's user avatar
  • 28.8k