On the number of classes of Gaussian genus whose arithmetic minimum is divisible by the square of a given odd number (Q1898270)

Let \(m > 0\), \(q > 0\) be integers with g.c.d. \((q,2m) = 1\) and \(({- m \over p}) = 1\) for all prime divisors of \(q\). Denote by \(h_1 (- m;G,q^2)\) the class number of integer-valued properly primitive positive binary quadratic forms of the genus \(G\) with determinant \(m\), whose arithmetic minimum is divisible by \(q^2\). The main result of the paper is the following: \[ h_1 (-m; G,q^2) \sim {2^{v(q)} \over \sigma_0 (q^2)} {h(-m) \over t} \quad \text{for} \quad m \to \infty \] where \(v(q)\) is the number of prime divisors of \(q\), \(\sigma_0 (q^2) = q^2 \prod_{p |q} (1 + {1 \over p})\), \(t\) is the number of the genera of determinant \(m\) and \(h(-m)\) is the class number of binary quadratic forms of determinant \(m\). The proof uses the discrete ergodic method introduced by Linnik. The Erratum in Math. Notes 55, No. 4, 435 (1994) supplies the corrected translation page 190.