ON THE NUMBER OF ORBITS ARISING FROM THE ACTION OF PSL(2, Z) ON IMAGINARY QUADRATIC NUMBER FIELDS

For square-free positive integers n, we study the action of the modular group PSL(2, Z) on the subsets { a+ root-n/c is an element of Q(root-n) vertical bar a, a(2)+n/c, c is an element of Z} of the imaginary quadratic number fields Q(root-n). In particular, we compute the number of orbits of this action and show, for n > 3, that it is equal to d(n) + 2/3 Sigma(n-1/2right perpendicular)(i=1left perpendicular ) [d(i(2) + n) - 2d(<=)i (i(2) + n)], where d(k) is the number of positive divisors of k, and d(<= i) (k) is the number of positive divisors of k which do not exceed i. We also provide a C++ code to calculate these numbers for square-free integers n with 1 <= n <= 100.