On the existence of unity in Lehmer's \(\psi\)-product ring
If F(x,y) is an integral-valued arithmetic function in two variables having the property that for every c the equation \(F(x,y)=c\) has only finitely many solutions, then Lehmer's product of two arithmetic functions f,g in one variable is defined by \(f*g(n)=\sum f(a)g(b)\), where the summation is taken over all pairs (a,b) of integers with \(F(a,b)=n\) [\textit{D. H. Lehmer}, Trans. Am. Math. Soc. 33, 945-957 (1931; Zbl 0003.10202); Am. Math. J. 53, 843-854 (1931; Zbl 0003.00303)]. The author shows that if F(x,y)\(\geq \max \{x,y\}\) and the set of all arithmetic functions with usual addition and Lehmer's product is a commutative ring then it has a unit element if and only if for every k the equation \(F(x,k)=k\) is solvable.
scientific article; zbMATH DE number 4144043