# Multiplicative property of representation numbers of ternary quadratic forms

Multiplicative property of representation numbers of ternary quadratic forms Let f be a positive definite integral ternary quadratic form and $$\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n$$ θ ( z ; f ) = ∑ n = 0 ∞ a ( n ; f ) q n its theta function. For any fixed square-free positive integer t with $$a(t;f)\ne 0$$ a ( t ; f ) ≠ 0 , we define $$\rho (n;t,f):=a(tn^2;f)/a(t;f)$$ ρ ( n ; t , f ) : = a ( t n 2 ; f ) / a ( t ; f ) . For the case when $$f=x_1^2+x_2^2+x_3^2$$ f = x 1 2 + x 2 2 + x 3 2 and $$t=1$$ t = 1 , Hurwitz proved that $$\rho (n;t,f)$$ ρ ( 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 $$\rho (n;t,f)$$ ρ ( n ; t , f ) for many cases. All cases in Cooper and Lam’s conjecture are included in ours. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

# Multiplicative property of representation numbers of ternary quadratic forms

Manuscripta Mathematica, Volume 156 (4) – Aug 21, 2017
11 pages

Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-017-0964-1
Publisher site
See Article on Publisher Site

### Abstract

Let f be a positive definite integral ternary quadratic form and $$\theta (z;f)=\sum _{n=0}^{\infty }a(n;f)q^n$$ θ ( z ; f ) = ∑ n = 0 ∞ a ( n ; f ) q n its theta function. For any fixed square-free positive integer t with $$a(t;f)\ne 0$$ a ( t ; f ) ≠ 0 , we define $$\rho (n;t,f):=a(tn^2;f)/a(t;f)$$ ρ ( n ; t , f ) : = a ( t n 2 ; f ) / a ( t ; f ) . For the case when $$f=x_1^2+x_2^2+x_3^2$$ f = x 1 2 + x 2 2 + x 3 2 and $$t=1$$ t = 1 , Hurwitz proved that $$\rho (n;t,f)$$ ρ ( 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 $$\rho (n;t,f)$$ ρ ( n ; t , f ) for many cases. All cases in Cooper and Lam’s conjecture are included in ours.

### Journal

Manuscripta MathematicaSpringer Journals

Published: Aug 21, 2017

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