Primitive Lattice-Point Densities for the Quadratic Forms in the Sieve of Atkin

We study primitive lattice-point activations associated with the three binary quadratic forms 4x² + y², 3x² + y², and 3x² − y² that underlie the sieve of Atkin. After translating the usual modulo-12 activation conditions into local restrictions on the coordinates and imposing gcd(x,y) = 1, we count admissible pairs (x,y) satisfying 0 < Q_i(x,y) ≤ N. Using lattice-point estimates with congruence conditions together with Möbius inversion, we obtain asymptotic formulas of the form E_i(N) = C_i N + O(sqrt(N) log N), with C_1 = 1/(2π), C_2 = sqrt(3)/(8π), and C_3 = [sqrt(3)/(4π²)] log(2 + sqrt(3)). Consequently, E(N) = C N + O(sqrt(N) log N), where C = 1/(2π) + sqrt(3)/(8π) + [sqrt(3)/(4π²)] log(2 + sqrt(3)) ≈ 0.2858504223. The logarithmic factor in the indefinite branch is identified with the regulator R_Q(sqrt(3)) = log(2 + sqrt(3)) of the real quadratic field Q(sqrt(3)). We also clarify the relation between the corresponding logarithmic coordinate and the action of the fundamental unit. Direct enumeration through N = 10^8 agrees closely with the predicted asymptotic constant.