The hexagonal versus the square lattice

A conjecture of Schmutz Schaller [17, p. 201] regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true. The proof uses results from (computational) prime number theory and from [10]. Using an identity due to Selberg, it is shown that the conjecture can in principle be also resolved without using computational prime number theory. By our approach, however, this would require a huge amount of computation.