On Gauss sums and the evaluation of Stechkin's constant (Q2814452)

The Gauss sums are defined by NEWLINE\[NEWLINE S_n\left(a,q\right)=\sum_{x\bmod q}\text{e}\left(\frac{ax^n}{q}\right), NEWLINE\]NEWLINE where \(\text{e}(t)=e^{2\pi it}\) for all \(t\in \mathbb{R}\). Denote NEWLINE\[NEWLINE A(n)=\sup_{q\geq 2}\max_{\text{gcd}(a,q)=1}\frac{\left|S_n\left(a,q\right)\right|}{q^{1-\frac{1}{n}}}. NEWLINE\]NEWLINE This paper shows that \(A(n)<A(6)\) for all \(n\geq 2\), \(n\neq 6\), and NEWLINE\[NEWLINE A(6)=\frac{\left|S_6\left(4787, 4606056\right)\right|}{4606056^{\frac{5}{6}}}=4.709236\dotsNEWLINE\]