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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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On Gauss sums and the evaluation of Stechkin’s constant
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by William D. Banks and Igor E. Shparlinski PDF
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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Additional Information
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 2569-2581
  • MSC (2010): Primary 11-XX
  • DOI: https://doi.org/10.1090/mcom3056
  • MathSciNet review: 3511293