Primes in Tuples of Linear Forms in Number Fields and Function Fields

TL;DR

This paper extends the Maynard-Tao theorem to admissible tuples of linear forms with arbitrary leading coefficients in number fields and function fields, broadening the scope of prime tuple results.

Contribution

It generalizes the Maynard-Tao theorem to include linear forms with arbitrary leading coefficients over number and function fields, beyond monic forms.

Findings

Established Maynard-Tao theorem for arbitrary leading coefficients in number fields.

Extended prime tuple results to function fields.

Provided applications of the generalized theorem.

Abstract

Following the work of Castillo-Hall-Oliver-Pollack-Thompson who extended Maynard-Tao theorem on admissible tuples to number fields and function fields for tuples with monic linear forms, here we obtain the Maynard-Tao theorem for admissible tuples of linear forms with arbitrary leading coefficients in number fields and function fields. Also, we provide some applications of our results.