Using Hooley neutralisers, a large sieve for quadratic characters is improved for multiplicatively weighted sequences, with applications to hyperbolic-region character sums.
Local solubility of generalised Fermat equations
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For every $n \geq 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $ax^n+by^n+cz^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations.
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An improved large sieve for quadratic characters via Hooley neutralisers and its applications
Using Hooley neutralisers, a large sieve for quadratic characters is improved for multiplicatively weighted sequences, with applications to hyperbolic-region character sums.