12
votes
1answer
695 views

Error to sum of Euler phi-functions

The number theory identity $\phi(1) + \phi(2) + \dots + \phi(n) \approx \frac{3n^2}{\pi^2}$ can be interpreted as counting relatively prime pairs of numbers $0 \leq \{ x,y \} \leq n$ . Has anyone ...
2
votes
0answers
125 views

finding rank-3 tensors compatible with a rank-2 tensor projection

I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any ...
4
votes
1answer
264 views

For an approach to the Hadamard-matrix-problem: is there a proof, that the iterative plane-wise orthogonal rotations (Quartimax/Varimax) converge to global maximum?

I've asked this question at stat-exchange and at the "Semnet"-mailing list of professionals in statistics. The reference to some articles in Psychometrica (for instance ten Berge 1995, Jennrich 2001) ...
7
votes
4answers
1k views

averages of Euler-phi function and similar

What are the odds two numbers are relatively prime? This is known to be $\frac{6}{\pi^2}$. The proof involves calculating averages of the Euler phi function. \[ \phi(1) + \phi(2) + \dots + \phi(n) ...
2
votes
1answer
490 views

Collatz conjecture and stationarity of time series

The Collatz conjecture is known to all. Has this question been approached by methods related to statistics? I think of Collatz iterates as a time series, and the question of whether we always get the ...
21
votes
2answers
915 views

Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all ...
0
votes
0answers
277 views

Estimating a multinomial sum

I have the following sum \begin{equation} \sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda} ...
0
votes
1answer
254 views

Estimating the probability of co-occurrence of a set of positive integers

How to estimate the probability of co-occurrence of the positive integers $c_i$ and $d_i$, $1 \leq i \leq t$ drawn from the uniform range $1$ to $2^k-1$, such that $\Sigma^t_{i=1} c^2_i = ...
8
votes
2answers
588 views

Notions of “independent” and “uncorrelated” for subsets of the natural numbers

In probability/statistics, there is a notion of two things being "independent", which would basically mean that any information we can get about one thing has no effect on our (probabilistic) ...