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Ternary quadratic forms with rational zeros. (English) Zbl 1219.11060

Let \(N(A, B)\) be the number of squarefree relatively prime integers \(a\leq A\), \(b\leq B\), for which the equation \(ax^2+by^2=z^2\) has a non-trivial rational solution. In the present work it is shown that for \(\delta>0\) and \(A, B\geq \exp((\log AB)^\delta)\) we have \[ N(A,B) = \frac{6}{\pi^3}\frac{AB}{\sqrt{\log A \log B}} \left\{1+\mathcal{O}\left(\frac{1}{\log A}+\frac{1}{\log B}\right)\right\}. \] A more complicated formula is shown for the case that \(b<(\log A)^C\) is fixed, and \(a\) varies up to \(A\). Further the number of integers \(a\) which satisfy only some of the local conditions is studied.
For the proof the quantity \(N(A,B)\) is expressed using the Hasse principle as a character sum. This results in a sum over quadratic characters weighted with the inverse over the number of divisors function, and having certain divisibility conditions among the summation conditions. For characters of small modulus the sum is evaluated using complex integration, while the contribution of characters of large modulus is bounded using a bilinear inequality.

MSC:

11E12 Quadratic forms over global rings and fields
11N25 Distribution of integers with specified multiplicative constraints
11N36 Applications of sieve methods
11L40 Estimates on character sums
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