The asymptotic distribution of exponential sums. II. (English) Zbl 1148.11043

Summary: 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 \(\sum_{c\leq X} S(f(x),c) \sim k(f)X^{4/3}\). 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.
In the case \(B=0\), an asymptotic formula was proved in a previous paper [Asian J. Math. 6, No. 4, 719–729 (2002; Zbl 1161.11371)] that contained a computational error. Now in Theorem 2.1 a corrected version of this result is given.
Part I, see Exp. Math. 12, No. 2, 135–153 (2003; Zbl 1061.11046).


11L07 Estimates on exponential sums
11Y35 Analytic computations
11N37 Asymptotic results on arithmetic functions
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