All Questions
Tagged with probabilistic-number-theory pr.probability
17 questions
-3
votes
0
answers
133
views
Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 619
1
vote
0
answers
78
views
In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?
Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements:
$\lambda$ being a random large prime such as $w^\lambda > 2\times m$
$1 < n < m−1$.
m is ...
- 87
4
votes
1
answer
239
views
Asymptotic density of an infinite union of subgroups
Let $1 < a_1 < a_2 < a_3 <{} ...$ be a sequence of integers. For a subset $A \subset \Bbb Z$, denote by $d(A)$ its natural density (if it exists).
Is it true that $$
\lim_{N \to +\infty} ...
- 255
1
vote
0
answers
123
views
On probability of coprimality of a list of numbers
We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
- 13.9k
2
votes
1
answer
138
views
Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$?
Let $s>1$ be a real number. We look at the zeta probability function / Zipf probability function defined as:
$$P(X = n) = \frac{1}{n^s \zeta(s)}$$
Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
- 3,032
2
votes
0
answers
98
views
Primes as expected values?
This is a follow-up question, which is related to the answer of this quesiton: Is there a connection of prime numbers and extreme value theory?
I will duplicate the answer here, so this question is ...
- 3,032
8
votes
2
answers
670
views
Ways of proving normal distribution (with a view towards Selberg's central limit theorem)
Given an random variable $Y:\Omega \to \mathbb{R}$ with finite mean $\mu$ and finite, positive variance $\sigma^2$, let $X = \frac{Y-\mu}{\sigma}$ be the renormalization with mean $0$ and variance $1$....
- 1,354
6
votes
1
answer
374
views
Almost evenly distributed spherical random vectors
Consider $n$ i.i.d spherically distributed random vectors $z_1 ,\cdots , z_n \sim \text{Unif}(\mathbb{S}^{d-1})$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ ...
- 315
2
votes
0
answers
170
views
The uniform “probability” on $\mathbf{N}$: What occurs beyond logarithmic density?
This is a follow-up to Question #47134. There is obviously no uniform probability distribution on $\mathbf{N}$ (or $\mathbf{Z}$); however, using the notion of amenability, you can show that any ...
- 284
0
votes
1
answer
138
views
Probabilistic interpretation of square free numbers and other properties
We can use the Lindberg condition to show the distribution of number of prime divisors of an integer approaches Gaussian.
Is there a similar probabilistic formulation for square free numbers? That is,...
- 1,826
3
votes
0
answers
237
views
Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)
Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...
- 10.5k
5
votes
1
answer
517
views
How many random matrices does it take to generate a matrix algebra?
Let $\mathbb{F}$ be a finite field.
Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra.
Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$
that one needs to take ...
- 3,104
15
votes
2
answers
10k
views
Convergence of moments implies convergence to normal distribution
I have a sequence $\{X_n\}$ of random variables supported on the real line, as well as a normally distributed random variable $X$ (whose mean and variance are known but irrelevant). I know that the ...
- 12.8k
3
votes
2
answers
462
views
using distribution of primes to generate random bits?
In his popular science book The Music of the Primes, Marcus du Sautoy tries to link the truth of the Riemann Hypothesis to the "randomness" of the primes. To do this, he invokes the idea of a "fair ...
- 371
24
votes
2
answers
1k
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 $...
- 551
26
votes
2
answers
2k
views
Is there any sense in which Dirichlet density is "optimal?"
A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| ...
- 19.2k
18
votes
1
answer
871
views
What's the probability that k + n^2 is squarefree, for fixed k?
While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is ...
- 14k