Multiplicative property of representation numbers of ternary quadratic forms

Let f be a positive definite integral ternary quadratic form and θ(z;f)=∑n=0∞a(n;f)qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n$$\end{document} its theta function. For any fixed square-free positive integer t with a(t;f)≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(t;f)\ne 0$$\end{document}, we define ρ(n;t,f):=a(tn2;f)/a(t;f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (n;t,f):=a(tn^2;f)/a(t;f)$$\end{document}. For the case when f=x12+x22+x32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=x_1^2+x_2^2+x_3^2$$\end{document} and t=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1$$\end{document}, Hurwitz proved that ρ(n;t,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (n;t,f)$$\end{document} 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 ρ(n;t,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (n;t,f)$$\end{document} for many cases. All cases in Cooper and Lam’s conjecture are included in ours.