TY - JOUR

T1 - On representations of integers by indefinite ternary quadratic forms

AU - Borovoi, Mikhail

N1 - Funding Information:
1Partially supported by the Hermann Minkowski Center for Geometry.

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - Ternary quadratic forms

UR - http://www.scopus.com/inward/record.url?scp=0035180727&partnerID=8YFLogxK

U2 - 10.1006/jnth.2001.2662

DO - 10.1006/jnth.2001.2662

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AN - SCOPUS:0035180727

SN - 0022-314X

VL - 90

SP - 281

EP - 293

JO - Journal of Number Theory

JF - Journal of Number Theory

IS - 2

ER -