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On Gauss sums and the evaluation of Stechkin's constant

Authors: William D. Banks and Igor E. Shparlinski
Journal: Math. Comp. 85 (2016), 2569-2581
MSC (2010): Primary 11-XX
Published electronically: November 20, 2015
MathSciNet review: 3511293
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Abstract: For the Gauss sums which are defined by

$\displaystyle S_n(a,q):=\sum _{x\bmod q}\mathbf {e}(ax^n/q), $

Stechkin (1975) conjectured that the quantity

$\displaystyle A:=\sup _{n,q\ge 2}~\max _{\gcd (a,q)=1}\frac {\bigl \vert S_n(a,q)\bigr \vert}{q^{1-1/n}} $

is finite. Shparlinski  (1991) proved that $ A$ is finite, but in the absence of effective bounds on the sums $ S_n(a,q)$ the precise determination of $ A$ has remained intractable for many years. Using recent work of Cochrane and Pinner (2011) on Gauss sums with prime moduli, in this paper we show that with the constant given by

$\displaystyle A=\bigl \vert S_6(\hat a,\hat q)\bigr \vert/\hat q^{1-1/6}=4.709236\ldots , $

where $ \hat a:=4787$ and $ \hat q:=4606056=2^3{\cdot }3^2{\cdot }7{\cdot }13{\cdot }19{\cdot }37$, one has the sharp inequality

$\displaystyle \bigl \vert S_n(a,q)\bigr \vert\le A\,q^{1-1/n} $

for all $ n,q\ge 2$ and all $ a\in \mathbb{Z}$ with $ \gcd (a,q)=1$. One interesting aspect of our method is that we apply effective lower bounds for the center density in the sphere packing problem due to Cohn and Elkies (2003) to derive new effective bounds on the sums $ S_n(a,q)$ in order to make the task computationally feasible.

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Additional Information

William D. Banks
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Igor E. Shparlinski
Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia

Received by editor(s): August 13, 2014
Received by editor(s) in revised form: March 7, 2015
Published electronically: November 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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