On representations of positive integers by $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$

TL;DR

This paper investigates the representation of positive integers using specific linear forms involving cube roots and linear combinations, employing sums of Liouville type to count the number of solutions under certain conditions.

Contribution

It introduces a method using sums of Liouville type to count representations of integers by complex forms involving cube roots and linear combinations.

Findings

Derived formulas for counting representations

Established conditions for the forms to represent integers

Extended previous work on integer representations

Abstract

We use sums of Liouville type to count the number of ways a positive integer can be represented by the forms $(a+c)^{1/3}x + (b+d)y$, $(a+c)x + \bigl(k(b+d) \bigr)^{1/3} y$, and $\bigl(k(a+c) \bigr)^{1/3} x + l(b+d) y$ for nonnegative integers $a,b,c,d,k,l,x,y$ under certain relative primality conditions.