The Fourier Coefficients of Θ‐series in Arithmetic Progressions

Hu, Guangwei; Jiang, Yujiao; Lü, Guangshi

doi:10.1112/mtk.12006

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2021

2023

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Cited by 8 publications

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“…en, we will use a new Voronoi-type summation formula generalized by Hu et al [13] to get the asymptotic formula, and this is the key to success. us, we can get the Kloosterman sum, use Weil's bound to get the saving in the q-aspect, and then obtain a similar main term as that in [12].…”

Section: Introductionmentioning

confidence: 99%

Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

Yan

2021

Journal of Mathematics

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Let r n denote the number of representations of a positive integer n as a sum of two squares, i.e., n = x 1 2 + x 2 2 , where x 1 and x 2 are integers. We study the behavior of the exponential sum twisted by r n over the arithmetic progressions ∑ n ∼ X n ≡ l mod q r n e α n β , where 0 ≠ α ∈ ℝ , 0 < β < 1 , e x = e 2 π i x , and n ∼ X means X < n ≤ 2 X . Here, X > 1 is a large parameter, 1 ≤ l ≤ q are integers, and l , q = 1 . We obtain the upper bounds in different situations.

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Section: Introductionmentioning

confidence: 99%