If the Ricci is a scalar multiple of the metric, does this imply that we also need torsion-free space? Since torsion is responsible for the anti-symmetric part of the connection, it would seem to me that is an additional requirement. Just confused about how torsion enters your calculation and if it would change anything in particular.

Thank you again, that helps! I think I understand from your calculation that the same should hold true even for curved space-time, although the example was done for d=3 curved space. Is this correct? Thank you again, I appreciate your help with this.

Thank you for that answer! Do you know of a good reference which shows the decomposition of the Reimann tensor as you've written it? I tried to show it using just the symmetries of the tensor but failed.

Turns out it's a fairly straightforward calculation - apply poisson summation on the function and then split the resulting sum over integers as $n = qr + k$ where $r\in\mathbb{Z}$ and $k = 0,1,\cdots q-1$. This gives the double sum and immediately leads to the desired result.