Let $rS^{d-1}$ denote the sphere of radius $r$ in dimension $d$ (centered at the origin). I'm interested in the number of lattice points on the sphere (not inside).

More precisely, let $$ N(r,d):=\text{number of lattice points on the sphere of raduis } r=\#\{x\in rS^{d-1}: x\in \mathbb{Z}^d\}. $$

I'm especially interested in the lower bound of $N(r,d)$ for any $d\ge 3$ and large $r$ (with $r^2\in\mathbb{Z}$, of course).

For example, I found in the book by F. Fricker *Einführung in die Gitterpunktlehre*. (German) [Introduction to lattice point theory] that the following result seems to be true (my German is poor):

$N(r,d)\gtrsim r^{d-2}$ for $d\ge 4$.

So what about $d=3$ case? What is the current best lower bound? The book is in 1982 so I guess there might be a better exponent than $d-2$ now.

One can also ask a weaker question: is there a sequence of $r$ tending to $\infty$ such that the above inequality holds with a better lower bound?