Representation of integers by positive ternary quadratic forms (a new modification of the discrete ergodic method)

One gives a detailed presentation of a new modification of the discrete ergodic method outlined in A. V. Malyshev's note (this Zapiski, 50, 179–186 (1975)). One gives new proofs for the asymptotic formulas obtained in Chap. VI of A. V. Malyshev's monography (Tr. Mat. Inst. Akad. Nauk SSSR, 65 (1962)). One obtains estimates for the remainder terms of this formulas under the assumption of some hypotheses about the zeros of Dirichlet's $L$-functions.