  We consider the combinatorial Laplacian on a sequence of discrete tori which
approximate the m-dimensional torus. In the special case m=1, Friedli and
Karlsson derived an asymptotic expansion of the corresponding spectral zeta
function in the critical strip, as the approximation parameter goes to
infinity. There, the authors have also formulated a conjecture on this
asymptotics, that is equivalent to the Riemann conjecture. In this paper,
inspired by the work of Friedli and Karlsson, we prove that a similar
asymptotic expansion holds for m=2. Similar argument applies to higher
dimensions as well. A conjecture on this asymptotics gives an equivalent
formulation of the Epstein-Riemann conjecture, if we replace the standard
discrete Laplacian with the $9$-point star discrete Laplacian.
