2023/04/20 by Naser T. Sardari, Sardari, Naser Talebizadeh, Orloski, Bryce Joseph
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2304.10021
openalex publication_date 2023/04/20 · openalex created_date 2023/04/24 · openalex updated_date 2026/07/28
Given a compact subset Σ⊂ ℝ (or ℂ) with logarithmic capacity greater than zero, we construct an explicit family of probability measures supported on Σ such that their closure is all the possible weak limit measures of complete sets of conjugate algebraic integers lying inside Σ. We give an asymptotic formula for the number of algebraic integers with given degree and prescribed distribution. We exploit the algorithmic nature of our approach to give a family of upper bounds that converges to the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the best previously known upper bound on the Schur-Siegel-Smyth trace problem to 1.8216.