On Gauss sums and the evaluation of Stechkin’s constant

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.