Eisenstein Series of 3/2 Weight and Eligible Numbers of Positive Definite Ternary Forms

A general algorithm is given for the number of representations for a positive integer n by the genus of a positive definite ternary quadratic form with form ax² + by² + cz². Using this algorithm, we study several nontrivial genera of positive ternary forms with small discriminants in the paper. As a conclusion we prove that f₁ = x² + y² + 7z² represents all eligible numbers congruent to 2 mod 3 except 14 _* 7^2k which was conjectured by Kaplansky in [K]. Our method is to use Eisenstein series of weight 3/2.