All Questions
Tagged with pr.probability nt.number-theory
36 questions
26
votes
5
answers
10k
views
Is there a simple way to compute the number of ways to write a positive integer as the sum of three squares?
It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 ...
- 14k
10
votes
2
answers
926
views
Isomorphisms between spaces of test functions and sequence spaces
I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
106
votes
5
answers
10k
views
integral of a "sin-omial" coefficients=binomial
I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof?
For any pair of integers $n\geq k\geq0$, we have
$$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...
- 43.2k
44
votes
5
answers
7k
views
Heuristically false conjectures
I was very surprised when I first encountered the Mertens conjecture. Define
$$ M(n) = \sum_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...
18
votes
3
answers
1k
views
Not-lonely runners
The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...
- 151k
3
votes
2
answers
973
views
Recursive random number generator based on irrational numbers
Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If $...
- 3,098
1
vote
1
answer
388
views
Curious inversion formula in additive combinatorics
Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions:
$N_S(z)$ is asymptotic continuous version of the function counting the ...
- 3,098
11
votes
4
answers
3k
views
If the sum of two independent random variables is discrete uniform on $\{a, \dots,a + n\}$, what do we know about $X$ and $Y$?
Basically I want to know whether the sum being discrete uniform effectively forces the two component random variables to also be uniform on their respective domains.
To be a bit more precise:
...
- 121
7
votes
1
answer
430
views
What makes Gaussian distributions special? Local field version?
This question is inspired by the recent one about Gaussian measures over the reals:
What makes Gaussian distributions special?
I would be interested in a similar list of characterizations for the ...
2
votes
1
answer
152
views
Computationally random bitstreams and normalcy
Let $\mathbb{N}$ denote the set of non-negative integers. We can identify every bitstream, i.e. a function $s:\mathbb{N}\to \{0,1\}$, with some $A\in{\cal P}(\mathbb{N})$: take $A = s^{-1}(\{1\})$.
...
- 48.9k
1
vote
2
answers
1k
views
Sum of digits iterated
Original version.
I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...
- 13.5k
66
votes
4
answers
4k
views
Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...
- 25.1k
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
16
votes
4
answers
1k
views
Random Diophantine polynomials: Percent solvable?
Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random polynomial:...
- 151k
15
votes
2
answers
5k
views
What areas of algebra could be interesting to probability theorists?
I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me ...
- 16.9k
15
votes
3
answers
2k
views
Probability that product is a perfect square
The probability a given integer in $[0,n]$ is a square is $\frac1{\sqrt n}$. What is the probability that if you take two integers uniformly then their product is square?
I know the main term is $\...
- 13.9k
14
votes
6
answers
2k
views
Density of numbers having large prime divisors (formalizing heuristic probability argument)
I want to prove that the set of natural numbers n having a prime divisor greater than $\sqrt{n}$ is positive.
I have a heuristic argument that this density should be $\log 2$, which is approximately ...
- 7,320
14
votes
1
answer
1k
views
Normal numbers, Liouville function, and the Riemann Hypothesis
This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...
- 3,098
14
votes
3
answers
1k
views
Probability of coprime polynomials
Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
- 54.2k
13
votes
1
answer
761
views
If $(a,b,c)$ are the sides of a triangle, then the probability $P(ax + by \ge c) = \frac{4}{\pi^2}\chi_2(x) + \frac{4}{\pi^2}\chi_2(y)$
Posting this question in MO since it is unanswered in MSE
Let $(a,b,c)$ be the side of a triangle. In its most general linear form, the triangle inequality can be expressed as: Does $ax + by \ge c$ ...
- 5,470
11
votes
2
answers
1k
views
Heuristic lower bounds on small sums of roots of unity
Let $f(k,n)$ be the smallest non-zero absolute value of a sum of $k$ complex $n$th roots of unity. Asking for bounds in either direction, Tao suggested that a polynomial lower bound seemed plausible ...
- 4,589
9
votes
2
answers
1k
views
Random pseudoprimes vs. primes
(Edit. What I called "pseudoprimes" are known as "Cramér random primes" in the literature,
of which I was unaware.)
Say that a set $S$ of natural numbers is a set of pseudoprimes if they
are (a) ...
- 151k
8
votes
2
answers
387
views
Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?
We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain.
Is it possible to estimate ...
user215601
8
votes
1
answer
380
views
Question about estimating random symmetric sums modulo p
Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
- 1,101
7
votes
2
answers
321
views
Random suborbits of a rotation
Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
- 2,560
7
votes
2
answers
639
views
Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
7
votes
0
answers
222
views
Projected polar chessboard measure convergence in total variation?
$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}\newcommand\ga{\gamma}$For natural $n$, let $E_n$ be the set of all points in $\R^2$ with "polar coordinates" $(r,t)$ in the set
$$F_n:=\...
- 128k
6
votes
3
answers
938
views
Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 61
5
votes
0
answers
614
views
is there a link with the probabilistic model for prime numbers?
Let $x \in \mathbb{R}_+$ and $k \in \mathbb{N}^{*}$.
Let :
$$\mathcal{A}(x)=\#\{(a_1, a_2, \ldots, a_k) \in \mathbb{P}^k \mid (a_1, a_2, \ldots, a_k \text{ verifying some properties}) \, , a_k \...
- 521
4
votes
1
answer
2k
views
Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?
A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...
3
votes
1
answer
315
views
Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)
If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
- 3,098
2
votes
1
answer
199
views
Average cluster size of a n-size vector
Given a vector of $n$ cells and $k$ elements in it, we can define a cluster of elements as a contiguous sequence of elements inside the vector.
My goal is to calculate the average cluster size for all ...
- 47
2
votes
2
answers
331
views
what's the best way to characterise the distribution of prime elements in simple perfect squared squares
DEFINITIONS: A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the ...
2
votes
1
answer
132
views
Independent decomposition of coordinate distribution
Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\...
- 515
2
votes
1
answer
386
views
How balanced can abc triples be?
I was looking at the $241$ known "good" abc triples (i.e. with quality $\geqslant1.4$), wondering how frequently $a$ and $b$ would have more or less the same order of magnitude. The outcome is not ...
- 13.4k
0
votes
0
answers
257
views
Unexpected autocorrelations in sequence of primes modulo 4
It is well known that there is a little bias in the distribution of prime residues modulo 4. But the bias eventually vanishes. I looked at the first million primes, and the counts are as follows:
...
- 3,098