Questions tagged [mobius-inversion]

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6
votes
1answer
367 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?
3
votes
1answer
306 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
0answers
115 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 ...
2
votes
0answers
125 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
3answers
351 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 ...
9
votes
2answers
359 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 ...
1
vote
1answer
247 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 ...
6
votes
3answers
789 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 $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
7
votes
0answers
769 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
2answers
549 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 ...
15
votes
2answers
551 views

Equivariant Mobius inversion

I'll first explain what Mobius 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 ...
2
votes
0answers
72 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
0answers
146 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, $...
0
votes
0answers
130 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)...
1
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0answers
94 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. ...
2
votes
0answers
231 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 ...
6
votes
4answers
794 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 ...
2
votes
0answers
341 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 $...
43
votes
2answers
8k 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
1answer
383 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 )^{...
1
vote
0answers
176 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)}...
1
vote
0answers
249 views

A conjecture on Moebius transformation

Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has n+3 distinct points any n+2 of which are in general position (in ...
4
votes
2answers
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)=[...
1
vote
1answer
581 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 ...
22
votes
2answers
2k 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....
13
votes
0answers
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 ...