On representations of integers by indefinite ternary quadratic forms

Abstract

Let $f$ be an indefinite ternary quadratic form, and let $q$ be an integer such that $-qdet(f)$ 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$tendstoinfinity,where$E(T,f,q)$istheHardy-Littlewoodexpectation(theproductoflocaldensities)and$0 \le c \le 2$.Wegiveexamplesof$f$and$q$suchthat$c$ takes the values 0, 1, 2.

AMSTeX, 10 pages