I'd like to know answers to the following.

In n-space, let I_n(R) denote the number of integer points lying on the sphere S_n(R) of radius R about the origin, and let f_n(R) denote the ratio I_n(R) / R^(n-1) (which for fixed n is the density #(integer points) / ((n-1)-area of S_n(R)), up to a constant factor).

QUESTIONS: For fixed n does f_n(R) reach a maximum over R for all n, some n, or no n ???

For any n for which it does, what is the maximum ovder R and for which R does it reach that maximum?

For n such that f_n(R) reaches no maximum, what is its supremum over R, and determine at least one sequence R_1, R_2, ... such that lim k -> oo (f_n(R_k)) = the supremum.

For starters, how about n = 2 and n = 3 ???

Note: The same density question has a mod p version as well for any dimension n. where the sphere of radius R is defined as all the points of (F_p)^n whose squares add up to R^2 mod p.  (In this case, of course, there are only a finite number of cases to consider, so the maximum density is always attained.)

In both the mod p and integer cases, of course only R = sqrt(integer) is relevant.

--Dan