On representations of integers by indefinite ternary quadratic forms

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Let f be an indefinite ternary integral quadratic form and let q be a nonzero 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 → ∞. We deduce from the results of our joint paper with Z. Rudnick that N(T, f, q) ∼ cEHL(T, f, q) as T → ∞, where EHL(T, f, q) is the Hardy-Littlewood expectation (the product of local densities) and 0≤c≤2. We give examples of f and q such that c takes the values 0, 1, 2.

Original languageEnglish
Pages (from-to)281-293
Number of pages13
JournalJournal of Number Theory
Issue number2
StatePublished - 2001


  • Ternary quadratic forms


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