TY - JOUR
T1 - SUMS OF KLOOSTERMAN SUMS OVER PRIMES IN AN ARITHMETIC PROGRESSION
AU - Dunn, Alexander
AU - Zaharescu, Alexandru
N1 - Publisher Copyright:
© 2018 The Author(s). All rights reserved.
PY - 2019/3/1
Y1 - 2019/3/1
N2 - For q prime, X≥1 and coprime u,v∈ℕ, we estimate the sums (Equation Presnted) where Kl 2 (p;q) denotes a normalized Kloosterman sum with modulus q. This is a sparse analog of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Milićević showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimization 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 Kiral, Petrow 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.
AB - For q prime, X≥1 and coprime u,v∈ℕ, we estimate the sums (Equation Presnted) where Kl 2 (p;q) denotes a normalized Kloosterman sum with modulus q. This is a sparse analog of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Milićević showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimization 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 Kiral, Petrow 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.
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U2 - 10.1093/qmath/hay035
DO - 10.1093/qmath/hay035
M3 - Article
AN - SCOPUS:85063775692
SN - 0033-5606
VL - 70
SP - 319
EP - 342
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
IS - 1
ER -