arXiv subjects

Connor Paul Wilson

Publications and source records attributed to Connor Paul Wilson.

4 recordsLinked to original sources

arXiv2021

On the Baire class of $n$-dimensional boundary functions

We show an extention of a theorem of Kaczynski to boundary functions in n-dimensional space. Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the $x$-axis. Suppose that $f$ is a function mapping $H$ into some metric space $Y$. If $E$ is any subset of $X$, we will say that a function $φ: E \rightarrow Y$ is a boundary function for $f$ if and only if for each $x\in E$ there exists an arc $γ$ at $x$ such that $\lim_{z\rightarrow x \atop z\inγ} f(z) = φ(x)$.

math.FA↗

arXiv2020

On coefficients satisfying Chebyshev's approximation of $π(x)$

We note an interesting and under-expressed fact from Chebyshev's initial bounding for the prime counting function, $π(x) := \# \{p \leq x : p \text{ prime}\},$ based upon a selection of fixed coefficients $d\in D$ to show $ψ(x) \asymp x$, and thus the goal of choosing some $a(d)$ approximately the same as $μ(d)$ such that: $$ \sum_{d}\frac{a(d)}{d} = 0, \quad \wedge \quad -\sum_{d}\frac{a(d)\log d}{d} \approx 1.$$

math.NT↗

arXiv2020

An improved bound on the sum-product estimate in $\mathbb{F}_{p}$

We give an improved bound on the famed sum-product estimate in a field of residue class modulo $p$ ($\mathbb{F}_{p}$) by Erdős and Szemeredi, and a non-empty set $A \subset \mathbb{F}_{p}$ such that: $$ \max \{|A+A|,|A A|\} \gg \min \left\{\frac{|A|^{15 / 14} \max \left\{1,|A|^{1 / 7} p^{-1 / 14}\right\}}{(\log |A|)^{2 / 7}}, \frac{|A|^{11 / 12} p^{1 / 12}}{(\log |A|)^{1 / 3}}\right\}, $$ and more importantly: $$\max \{|A+A|,|A A|\} \gg \frac{|A|^{15 / 14}}{(\log |A|)^{2 / 7}}.$$

math.CO↗

arXiv2020

Estimates of the bounds of $π(x)$ and $π((x+1)^{2})-π(x^{2})$

We show the following bounds on the prime counting function $π(x)$ using principles from analytic number theory, giving an estimate: $$2 \log 2 \geq \limsup_{x \rightarrow \infty} \frac{π(x)}{x / \log x} \geq \liminf_{x \rightarrow \infty} \frac{π(x)}{x / \log x} \geq \log 2$$ for all $x$ sufficiently large. We also conjecture about the bounding of $π((x+1)^{2}) - π(x^{2})$, as is relevant to Legendre's conjecture about the number of primes in the aforementioned interval such that: $$ \left \lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log x}\right)-\frac{\left(\log x\right)^{2}}{\log\left(\log x\right)}\right \rfloor \leq π((x+1)^{2}) - π(x^{2}) \leq $$ $$ \left \lfloor\frac{1}{2}\left(\frac{\left(x+1\right)^{2}}{\log\left(x+1\right)}-\frac{x^{2}}{\log x}\right) + \log^{2}x\log\log x \right \rfloor$$

math.NT↗