arXiv subjects

Rodrigo Angelo

Publications and source records attributed to Rodrigo Angelo.

5 recordsLinked to original sources

arXiv2024

Oscillations of random multiplicative functions under initial bias

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-\epsilon}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow \infty$. This solves a conjecture of Kucheriaviy, who has a complementary result showing this exponent is sharp. We also prove that almost surely the partial sums of $\sum\frac{f(n)}{\sqrt{n}}$ change signs infinitely many times, solving a problem of Aymone.

math.NT↗

arXiv2022

A polynomial with a root mod $p$ for every $p$ has a real root

We prove that if a polynomial has a root mod $p$ for every large prime $p$, then it has a real root. As an application, we show that the primes can't be covered by finitely many positive definite binary quadratic forms.

math.NT↗

arXiv2022

A hidden signal in Hofstadter's $H$ sequence

The Hofstadter $H$ sequence is defined by $H(1) = 1$ and $H(n) = n-H(H(H(n-1)))$ for $n > 1$. If $\alpha$ is the real root of $x^3+x=1$ we show that the numbers $\alpha H(n) \mod 1$ are not uniformly distributed on $[0,1]$, but converge to a distribution we believe is continuous but not differentiable. This is motivated by a discovery of Steinerberger, who found a real number with similar behavior for the Ulam sequence. Our result is related with the fact that a certain sequence defined from the linear recurrence $h_n=h_{n-1}+h_{n-3}$ has the property $\|x h_n\| \rightarrow 0$ precisely for $x \in \mathbb{Z}[\alpha]$, a phenomenon we inquire for general linear recurrent sequences of integers.

math.NT↗

arXiv2022

On a Tur\'an conjecture and random multiplicative functions

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.

math.NT↗

arXiv2017

A Cameron and Erd\"os conjecture on counting primitive sets

Let $f(n)$ count the number of subsets of $\{1,...,n\}$ without an element dividing another. In this paper I show that $f(n)$ grows like the $n$-th power of some real number, in the sense that $\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists. This confirms a conjecture of Cameron and Erd\"os, proposed in a paper where they studied a number of similar problems, including the well known "Cameron-Erd\"os os Conjecture" on counting sum-free subsets.

math.NT↗