Notes on two lemmas concerning the Epstein zeta-function

We use Cassels's notation and define h ( m, n ), Q ( m, n ), Z h ( s ), Z h (1) – Z Q (1) and G ( x, y ) as in [1]. Rankin [5] proved that the Epstein zeta-function Z h ( s ) satisfies, for s ≧ 1·035, the THEOREM. For s > 0, Z h ( s ) — Z Q ( s ) ≧ 0 with equality if and only ifh is equivalent to Q . Rankin then asked whether the theorem is true for all s > 1. Cassels [1] answered this question in the affirmative and proved further that the theorem is true for all s > 0.