Let $$ (\log x_1)^2+ (\log x_2)^2 + \cdot\cdot\cdot + (\log x_n)^2= r $$
For $r=0,1,2,…$
The goal is to count the number of lattice points enclosed by the curve/surface for $r$. In this way the problem is a generalization of the Gauss circle problem.
What is the exact solution or at least an approximate solution in dimension $2$?
I calculated that the first several terms in the sequence are $1,4,10,15,28,42,…$
The sequence seems to grow more slowly than when compared to the usual Gauss circle problem.
First of all, I feel the necessity to represent some of these curves :
Fig. 1 : the first 5 curves for $r=1,\cdots 5$. In fact the "curve" for $r=0$ is reduced... to a single point $(1,1)$.
The asymptotic growth of these numbers $$N(r)=1,4,10,15,28,42,59,86,111,152,197,252,319,404,504,621,752,916,1112,1331,1589,1889,2233,2629,3081,3596,4192,4865,5634,6501,7480\cdots$$ look, on a "trial and error" basis to be approximately governed (on a logarithmic scale) by the following approximation :
$$\ln(N(r)) \approx 1.73r^{0.48}\tag{1}$$
(as represented on the figure below) :
Otherwise said :
$$N(r) \approx \exp(1.73r^{0.48})$$
Fig. 2 : Representation on a log. scale : exact figures (function $\log(N(r))$ represented as the blue curve, approximate values (RHS of (1)) represented by the red curve.