There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field.

That number has a natural geometric approximation $G_a$, and we call the difference $E_a=N_a-G_a$ "the error term". The partial sum of the error term $\sum_{a<X}E_a$ is a classical object and the big O result is studied. I wonder whether there are big Omega results about it. (i.e. some lower bound of its order when $X\rightarrow \infty$.)