ON SHORT SUMS OF TRACE FUNCTIONS

被引：19

作者：

Fouvry, Etienne ^{[1
]}

Kowalski, Emmanuel ^{[2
]}

Michel, Philippe ^{[3
]}

Raju, Chandra Sekhar ^{[4
]}

Rivat, Joel ^{[5
]}

Soundararajan, Kannan ^{[4
]}

机构：

[1] Univ Paris Saclay, Univ Paris Sud, CNRS, Lab Math Orsay, F-91405 Orsay, France

[2] ETH, D MATH, Razmistr 101, CH-8092 Zurich, Switzerland

[3] Ecole Polytech Fed Lausanne, Mathgeom TAN, Stn 8, CH-1015 Lausanne, Switzerland

[4] Dept Math, 450 Serra Mall, Stanford, CA 94305 USA

[5] Univ Aix Marseille, Inst Math Marseille, Case 907, 163 Ave Luminy, F-13288 Marseille 9, France

来源：

ANNALES DE L INSTITUT FOURIER | 2017年 / 67卷 / 01期

基金：

美国国家科学基金会; 奥地利科学基金会; 瑞士国家科学基金会;

关键词：

Short exponential sums; trace functions; van der Corput lemma; completion method; Riemann Hypothesis over finite fields;

D O I：

10.5802/aif.3087

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Polya-Vinogradov gap" in some sense) for bounded functions with bounded Fourier transforms. We then prove that the existence of non-trivial estimates for ranges slightly below the square-root bound is stable under the discrete Fourier transform. We then give applications related to trace functions over finite fields.

引用

页码：423 / 449

页数：27

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