Multiplicative property of representation numbers of ternary quadratic forms

Abstract

Let f be a positive definite integral ternary quadratic form and $$\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n$$ its theta function. For any fixed square-free positive integer t with $$a(t;f)\ne 0$$ , we define $$\rho (n;t,f):=a(tn^2;f)/a(t;f)$$ . For the case when $$f=x_1^2+x_2^2+x_3^2$$ and $$t=1$$ , Hurwitz proved that $$\rho (n;t,f)$$ is multiplicative and he gave its expression. Cooper and Lam proved four similar formulas and proposed a conjecture for some other cases. Using the results given in this paper, we can check the multiplicative property of $$\rho (n;t,f)$$ for many cases. All cases in Cooper and Lam’s conjecture are included in ours.