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 $\bigl(k(a+c) \bigr)^{1/3} x +   l(b+d) y$
  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. 
Mohamed El Bachraoui
math.NT