SUMS OF KLOOSTERMAN SUMS OVER PRIMES IN AN ARITHMETIC PROGRESSION

@article{Dunn2019SUMSOK,
  title={SUMS OF KLOOSTERMAN SUMS OVER PRIMES IN AN ARITHMETIC PROGRESSION},
  author={Alexander Dunn and Alexandru Zaharescu},
  journal={The Quarterly Journal of Mathematics},
  year={2019}
}
For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where $\text{Kl}_2(p;q)$ denotes a normalised Kloosterman sum with modulus $q$. This is a sparse analogue of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Mili\'cevi\'c showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use… 
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