Hybrid bounds for quadratic Weyl sums and arithmetic applications

@article{Liu2015HybridBF,
  title={Hybrid bounds for quadratic Weyl sums and arithmetic applications},
  author={Sheng-chi Liu and Riad Masri},
  journal={Forum Mathematicum},
  year={2015},
  volume={27},
  pages={3397-3423}
}
Let D < 0 be an odd fundamental discriminant and q be a prime number which splits in Q( √ D). Given a suitable smooth function f supported on [X, 2X] for X ≥ 1, we establish a uniform bound in X,D and q for ∑ c≡0 (mod q) Wh(D; c)f(c), 

On the distribution of modular square roots of primes

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be

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