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Fourier optimization and quadratic forms

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arxiv 2012.07781 v2 pith:6HNW7OE5 submitted 2020-12-14 math.NT math.CA

Fourier optimization and quadratic forms

classification math.NT math.CA
keywords quadraticdefiniteformsfourierintegerspositiverepresentedresults
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We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\geq 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun-Titchmarsh-type results in short intervals, and a conditional Cram\'er-type result on the maximum gap between primes represented by a given positive definite quadratic form.

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