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

Abstract

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

arXiv commentthese results have been published in another paper of the author