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### Ratio of consecutive divisors and average

Let $2\leq d_1 < d_2,...,d_l < n$ be all the proper nontrivial divisors of $n$. I like to understand how much these divisors deviates from each other. Here are two questions in this regard:
(1) ...

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### Distribution function for divisors of an Integer

For a fixed $n$, let $D_n(x) = \{ d|n : d \leq x \}$ . We assume here $p \leq x \leq n/p$,
where $p$ is the smallest prime factor of $n$.
For example if $n = p^i$ for some prime $p$ then $D_n(x) ...

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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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### 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 ...

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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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### 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 ...

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### 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]| ...

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### Coprime polynomials

Consider the set of polynomials (in standard form) P: the coefficients in the polynomials are integers.
Quite like Z (integers), every distinct member of P has a unique 'prime factorization'. ...

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### 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 ...

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### U(3) Sato-Tate measure.

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 ...

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### 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 ...