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Abstract
For the Gauss sums which are defined by [equation presented here], Stechkin (1975) conjectured that the quantity [equation presented here] 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 = S_{6}(â,qˆ)/qˆ^{1} ^{1/6 }=4.709236...,
where â := 4787 and qˆ := 4606056 = 2^{3}·3^{2}·7·13·19·37, one has the sharp inequality
Sn(a,q) ≤ Aq^{1} ^{1/n}
for all n, q ≥ 2 and all a ε 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.
Original language  English 

Pages (fromto)  25692581 
Number of pages  13 
Journal  Mathematics of Computation 
Volume  85 
Issue number  301 
DOIs  
Publication status  Published  2016 
Externally published  Yes 
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Projects
 1 Finished

New Applications of Additive Combinatorics in Number Theory and Graph Theory
Mans, B., Shparlinski, I., MQRES, M. & PhD Contribution (ARC), P. C. (.
1/01/14 → 31/12/17
Project: Research