Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields

TL;DR

This paper extends classical Mertens and Mirsky formulas to the Euler functions of principal rings of integers in imaginary quadratic number fields, providing asymptotic behaviors in sectors and with congruences.

Contribution

It introduces new asymptotic formulas for Euler functions in imaginary quadratic fields, generalizing classical results to more complex algebraic settings.

Findings

Derived sectorial asymptotic formulas for Euler functions

Extended Mirsky and Mertens formulas to imaginary quadratic fields

Provided congruence-based versions of the formulas

Abstract

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the standard Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sector and with congruences.