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- W2962879303 abstract "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 an optimisation argument inspired by Fouvry, Kowalski and Michel. Our argument compares three different bounds for bilinear forms involving Kloosterman sums. The first input in this method is a bilinear bound we prove using uniform asymptotics for oscillatory integrals due to Petrow, Kiral and Young. In contrast with the case when the sum runs over all primes, we exploit cancellation over a sum of stationary phase integrals that result from a Voronoi type summation. The second and third inputs are deep bilinear bounds for Kloosterman sums due to Fouvry-Kowalski-Michel and Kowalski-Michel-Sawin." @default.
- W2962879303 created "2019-07-30" @default.
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- W2962879303 date "2019-03-01" @default.
- W2962879303 modified "2023-09-23" @default.
- W2962879303 title "SUMS OF KLOOSTERMAN SUMS OVER PRIMES IN AN ARITHMETIC PROGRESSION" @default.
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- W2962879303 doi "https://doi.org/10.1093/qmath/hay035" @default.
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