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? 



Lower bound for the number of lattice points on high dimensional spheres
My answer to [this MO question](https://mathoverflow.net/questions/217698/many-representations-as-a-sum-of-three-squares) contains the answer to your question, especially if you take into account that $L\left(1,\left(\frac{D}{\cdot}\right)\right)$ can be estimated unconditionally (i.e. without GRH):
$$|D|^{-\varepsilon}\ll\_\varepsilon L\left(1,\left(\tfrac{D}{\cdot}\right)\right)\ll \log|D|.$$
The lower bound is ineffective (i.e. we don't know the implied constant), and it is due to Siegel (1934). The upper bound is effective and older. Both bounds are explained in Montgomery-Vaughan: Multiplicative number theory I.


