On representations of integers by indefinite ternary quadratic forms

Details On representations of integers by indefinite ternary quadratic forms On representations of integers by indefinite ternary quadratic forms

2000-06-20

arxiv.org/abs/math/0006141v1

Mathematics

Number Theory

AMSTeX, 10 pages

Scientific paper

Let $f$ be an indefinite ternary quadratic form, and let $q$ be an integer such that $-q det(f)$ is not a square. Let $N(T,f,q)$ denote the number of integral solutions of the equation $f(x)=q$ where $x$ lies in the ball of radius $T$ centered at the origin. We are interested in the asymptotic behavior of $N(T,f,q)$ as $T$ tends to infinity. We deduce from the results of our joint paper with Z. Rudnick that $N(T,f,q)$ grows like cE(T,f,q)$ as $T$ tends to infinity, where $E(T,f,q)$ is the Hardy-Littlewood expectation (the product of local densities) and $0 \le c \le 2$. We give examples of $f$ and $q$ such that $c$ takes the values 0, 1, 2.

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