Partition regularity in imaginary quadratic rings of integers

TL;DR

This paper investigates partition regularity of quadratic equations in imaginary quadratic integer rings, introducing new number-theoretic results and tools for analyzing solutions over these algebraic structures.

Contribution

It develops novel number-theoretic techniques and characterizations for imaginary quadratic rings, advancing understanding of partition regularity in these algebraic settings.

Findings

Established partition regularity results for certain quadratic equations

Developed a characterization for aperiodic completely multiplicative functions

Proved a new concentration estimate for multiplicative functions

Abstract

We obtain partition regularity results for homogeneous quadratic equations whose parametrized solutions admit nice factorizations into linear forms over rings of integers of imaginary quadratic fields. To do so, we develop number-theoretic results of independent interest on such fields, such as a characterization for aperiodic completely multiplicative functions, the Tur\'an-Kubilius inequality, and a new concentration estimate for multiplicative functions.