# Questions tagged [probabilistic-number-theory]

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### Small linear relations between primitive Pythagorean triples $\mathsf{II}$

WillJagy answered a linear relation question on Pythagorean Triples in Small linear relations between primitive Pythagorean triples $\mathsf I$. Now let $a^2+b^2=c^2$ be a primitive Pythagorean ...
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### What is the probability of 'yes' to this likely $coNP$ problem?

Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$. Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...
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### Probability of small solutions to an uniform random linear diophantine equation?

Take the set $$T(c_1,\dots,c_t)=\{(x_1,\dots,x_t)\in\mathbb Z^t\backslash\{(0,\dots,0)\}:\sum_{i=1}^tc_ix_i\equiv0\bmod q\}$$ where $c_1,\dots,c_t\in(-q/2,q/2)\cap\mathbb Z$. What is probability ...
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### How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?

Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
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### expected number of shared 1s between two binary strings from a given set

Let say, I have two binary strings with length N, chosen from a set where there are $2^N-K,(K \ge 0)$ independent strings. What would be the expected number of Ones at the same index from two randomly ...
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### 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 ...
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### Relation between the densities of two functions and their additive convolution?

Suppose that for two integral-valued arithmetic functions $f_i$ ($i=1,2$), the following values are known: $$\lim_{n\to \infty} \frac{ \{n:f_i(n) \text{ is odd}\}\cap \{0,1,\ldots, n-1\} }{n}.$$ (In ...
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### 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 ...
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### Has every divisibility-antichain density zero?

Let $A \subset \mathbb N$ be a antichain with respect to divisibility. Does this imply that the density of $A$ is $0$?
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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 $... 1answer 400 views ### Work exploring application of probability to metric number theory problems I am interested in studying the application of probabilistic tools to study metric number theoretic problems, specifically the Duffin-Schaeffer conjecture (http://www.math.osu.edu/files/duffin-... 2answers 2k views ### Is there any sense in which Dirichlet density is “optimal?” A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| ... 1answer 1k views ### Heuristic reason for Polya's conjecture Let$\lambda(n)$be Liouville's function, so that for each positive integer$n = p_1^{m_1}\cdots p_r^{m_r}$, we have that$\lambda(n) = (-1)^{\sum^{r}_{k=1}{m_k}}$. In 1919, Polya conjectured that$L(...
An undergraduate is performing some computations, related to a Sato-Tate conjecture of $U(3)$ type (a curve over $Q$, for which the roots of local L-functions look like eigenvalues of a random matrix ...