# Questions tagged [mobius-inversion]

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19
questions

**9**

votes

**0**answers

596 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 }...

**0**

votes

**0**answers

59 views

### Prove that the Mertens function is invariant under matrix inversion for $n>3$

Consider the lower triangular matrix $T$ with the definition:
$n \leq3 :$
$$T(n, 1)=1$$
$n>3:$
$$T(n,1)=x_n$$
$n \geq k:$
$n \leq 3:$
$$T(n,k)=1$$
$n \geq k:$
$n>3:$
$k=2 \text{ or } k=3:$
...

**6**

votes

**2**answers

427 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

**2**answers

506 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

**0**answers

63 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

144 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

**0**answers

125 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**

vote

**0**answers

93 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

**0**answers

217 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. (...

**6**

votes

**4**answers

643 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

**0**answers

309 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 $...

**40**

votes

**2**answers

7k 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

365 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

**0**answers

174 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

**0**answers

248 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

**2**answers

973 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

**1**answer

528 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 ...

**21**

votes

**2**answers

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

**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 ...