Exponential sum bound of Mordell and Hua
Volume 218 / 2025
Abstract
We improve estimates for the exponential sum $ S(f,q)=\sum _{x=1}^q e_q(f(x)), $ where $e_q(\cdot )=e^{2\pi i\, \cdot /q}$ and $f(x)$ is a primitive polynomial over $\mathbb Z$. Let $R(f,q)= |S(f,q)|/q^{1-1/k}$, with $k$ the degree of $f$, and $R(k,q)$ be the maximum value of $R(f,q)$ over primitive polynomials of degree $k$. Among other results, we show that for any prime power $p^m$ with $5 \le p \le 2k-1$ we have $ R(k,p^m) \le p^{\frac 2{p+1}+\frac 1p}. $ In particular, $R(k,p^m) \le 2.815$ for any $k$ and prime power $p^m$. We also show that for any positive integer $q$, $$ R(k,q) \le e^{k+\frac 1{2\pi}\log^2k+6\log k -4.88891} $$ for $k \lt 4.62 \cdot 10^{12}$ unconditionally, and for all $k \ge 1$ on the assumption of the Riemann Hypothesis. Refined estimates are given for small $k$.
