Visual properties of generalized Kloosterman sums

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• 作者：Paula Burkhardt^b ; Alice Zhuo-Yu Chan^a ; ^{azchan@ucsd.edu" class="auth_mail" title="E-mail the corresponding author} ; Gabriel Currier^b ; Stephan Ramon Garcia^b ; ^{Stephan.Garcia@pomona.edu" class="auth_mail" title="E-mail the corresponding author} ; Florian Luca^c ; ^{Florian.Luca@wits.ac.za" class="auth_mail" title="E-mail the corresponding author} ; Hong Suh^b
• 关键词：Kloosterman sum ; Gauss sum ; Salié ; sum ; Supercharacter ; Hypocycloid ; Uniform distribution ; Equidistribution ; Lucas number ; Lucas prime
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：237-253
• 全文大小：4560 K

文摘

For a positive integer m   and a subgroup Λ of the unit group class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002875&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002875&_rdoc=1&_issn=0022314X&md5=6576de7faff52f52ac886aa9700dae94" title="Click to view the MathML source">(Z/mZ)^×class="mathContainer hidden">class="mathCode">${(\mathbb{Z}/m\mathbb{Z})}^{\times }$, the corresponding generalized Kloosterman sum   is the function class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002875&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002875&_rdoc=1&_issn=0022314X&md5=8bcf231564dd813c293c078be63ea2d8">class="imgLazyJSB inlineImage" height="23" width="234" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002875-si2.gif">class="mathContainer hidden">class="mathCode">$K(a,b,m,\mathrm{\Lambda })={\sum }_{u\in \mathrm{\Lambda }}e(\frac{au+b{u}^{-1}}{m})$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002875&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002875&_rdoc=1&_issn=0022314X&md5=eeb7fe6de3a2cf39cac8a5b01ee91144" title="Click to view the MathML source">a,b∈Z/mZclass="mathContainer hidden">class="mathCode">$a,b\in \mathbb{Z}/m\mathbb{Z}$. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.