On a class of enumeration problems in additive arithmetics (Q1310893)

Given a linear form \(\sum_ i h_ i x_ i\) in \(k\) variables with \(h_ i\in \mathbb{N}\), the author computes \[ \#\{ (x_ i)\in \mathbb{N}^ k:\;1\leq x_ i\leq r,\;(x_ i,r)=1, (\sum h_ i x_ i,r) =1\}, \] that is, the number of \(k\)-dimensional points with natural coordinates \(\leq r\) and relatively prime to \(r\), which give function values also prime to \(r\). In a second theorem he changes the conditions \((x_ i, r)=1\) to \((x_ i,r)>1\), \(1\leq i\leq k\). In his computations he uses Ramanujan sums.