The Asymptotic Distribution of Exponential Sums, II

Abstract

Let $f(x)$ be a polynomial with integral coefficients and let, for $c>0$, $S(f(x),c)=\sum_{j \!\! \pmod c} \exp(2\pi\imath\frac{f(j)}c)$. If $f$ is a cubic polynomial then it is expected that $\sum_{c\le X} S(f(x),c) \sim k(f)X^{4/3}$. In this paper, we consider the special case $f(x)=Ax^3+Bx$ and propose a precise formula for $k(f)$. This conjecture represents a refined version of the classical Kummer conjecture.