Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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On Gauss sums and the evaluation of Stechkin’s constantHTML articles powered by AMS MathViewer

by William D. Banks and Igor E. Shparlinski
Math. Comp. 85 (2016), 2569-2581 Request permission

Abstract:

For the Gauss sums which are defined by $S_n(a,q):=\sum _{x\bmod q}\mathbf {e}(ax^n/q),$ Stechkin (1975) conjectured that the quantity $A:=\sup _{n,q\geqslant 2}~\max _{\gcd (a,q)=1}\frac {\bigl |S_n(a,q)\bigr |}{q^{1-1/n}}$ 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 $A=\bigl |S_6(\hat a,\hat q)\bigr |/\hat q^{1-1/6}=4.709236\ldots ,$ 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 $\bigl |S_n(a,q)\bigr |\leqslant A q^{1-1/n}$ for all $n,q\geqslant 2$ 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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• William D. Banks
• Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
• MR Author ID: 336964
• Email: bankswd@missouri.edu
• Igor E. Shparlinski
• Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
• MR Author ID: 192194
• Email: igor.shparlinski@unsw.edu.au
• Received by editor(s): August 13, 2014
• Received by editor(s) in revised form: March 7, 2015
• Published electronically: November 20, 2015