All Questions
23 questions
18
votes
3
answers
918
views
Can Gauss sums derandomize any heuristic arguments?
I've always been fascinated by the fact that the classical Gauss sum has absolute value $\sqrt p$, which is exactly what we would expect if we were to interpret the Gauss sum as a random walk. In ...
- 82.7k
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
13
votes
4
answers
1k
views
What results would follow from or imply "randomness" of the primes?
This question on random versions of deterministic problems reminded me that many conditional results in number theory hold if the primes are in some sense random, and it is common knowledge that the ...
9
votes
6
answers
3k
views
Primes are pseudorandom?
I've been reading the wonderful slides by Terry Tao and thought about this question.
Primes appear to be quite random, and the formal statement should be that there are some characteristics of primes ...
- 15.1k
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
7
votes
0
answers
267
views
Can primes be (almost) random sequence in von Mises sense?
Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
- 7,182
6
votes
4
answers
2k
views
Probability that randomly chosen integers from a restricted set of natural numbers are coprime
We know that the probability $P(k)$ of $k$ randomly chosen integers $(k \ge 2)$ from the set of natural number are coprime is
$$
P(k) = \frac{1}{\zeta(k)}.
$$
I am looking at a special case of ...
- 5,470
6
votes
1
answer
481
views
Probabilistic Proofs of Key Number-Theoretic Results
Given a positive integer $n$, let $p$ be the largest prime less than or equal to $n$.
Let $N(n)=2^{C_2}\cdots p^{C_p}$ be uniformly distributed from $1$ to $n$, and $M(n)=2^{Z_2}\cdots p^{Z_p}$ where ...
- 329
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
0
answers
536
views
Is the integer factorization into prime numbers normally distributed?
Edit:
Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking.
Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, ...
- 3,064
3
votes
1
answer
401
views
Probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$
Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ ...
- 1,776
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
3
votes
0
answers
286
views
What is the value of this simple game with primes?
Consider the following game. Alice selects an integer $n$ from $[1,b]$, while Bob selects an integer $m$ from $(a,b]$ (for concreteness, you may choose $a=10^{10}$ and $b=10^{1000}$). Alice wins if $m-...
- 781
2
votes
1
answer
159
views
Is the density of integers $n$ such that the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ is surjective positive?
Let $\omega(m)$ be the number of prime factors of $m$ regardless of multiplicity. I'm interested in the behavior of the finite sequence $(\omega(n-r)\omega(n+r))_{0\leq r\leq n-1}$ for a given integer ...
- 6,984
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 ...
- 3,064
2
votes
0
answers
87
views
The covariance of certain random variable
We define two random variables $X_n,Y_n $ on the sample space $\{1,2,3,\cdots,n\}$ with counting measure. We denote by $C_n$ the covariance of theses two random variables: $C_n=Cov(X_n,Y_n)$.
...
- 356
1
vote
2
answers
513
views
Primes as uncorrelated random variables [closed]
The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that
the number of twin primes below $x$ should be roughly $\dfrac{x}{\...
- 6,984
1
vote
1
answer
390
views
Probability that p and q are both prime provided q-p=2r
Hello,
I would like to know whether there is a way, thanks to the prime number theorem, to give some kind of an equivalent of the probability that two positive integers $p$ and $q$ less than a given ...
- 6,984
1
vote
1
answer
199
views
Asymptotic for the probability that a number has $k$ prime factors less than $Q$
If we let $\omega_Q(n)$ denote the number of distinct prime factors of $n$ less than a bound $Q$, then what asymptotic formulas exist for $\Pr_{n\in\mathbb{N}}[\omega_Q(n)=k]$ as $Q\to\infty$ if $k$ ...
- 2,902
1
vote
0
answers
641
views
Some stuff related to the twin prime conjecture
I am making three statements here, and my question is about statement 2, asking if someone can prove or disprove it. A (possibly weaker) version of statement 2 was proved as an answer to a former ...
- 3,098
0
votes
0
answers
102
views
Formalizing the "pseudorandomness" of primes
Many conjectures about primes seem to revolve around the idea of "primes are random". So I thought about how this "randomness" may be formally defined, and came up with the ...
- 1
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