Work

On extensions of Dirichlet and Green–Tao theorems and Goldbach–Dirichlet representations over certain families of commutative rings with unity

Year: 2025

Type: article

Abstract: In this paper, we study stronger forms of Goldbach's conjecture enriched with the linear representations of prime numbers given by the classical Dirichlet theorem and its extensions. We call such a representation a Goldbach–Dirichlet representation (GD‐representation). Among other results, we show that Dirichlet's theorem on arithmetic progressions is, in general, not true in the ring of formal power series over the integers. Additionally, we use a polynomial version of the Schinzel hypothesis due to A. Bodin, P. Dèbes, and S. Najib to prove the existence of GD‐representations for a wide collection of polynomial rings over special families of fields of characteristic zero, among others. Moreover, we study the (non)validity of Dirichlet's theorem over several families of commutative rings with unity like polynomial and formal series rings. Finally, we obtain a generalization for polyomial rings of the celebrated Green–Tao theorem.

Source: Mathematische Nachrichten

DOI: 10.1002/mana.70080

Authors: Danny A. J. Gómez‐Ramírez, Alberto F. Boix

Institution: Universitat Politècnica de Catalunya

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Ministerio de Ciencia, Innovación y Universidades

Award: G3733237515