## 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).

### MSC:

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

### Keywords:

Cubic exponential sums; Kummer conjecture; Gauss sums

### Citations:

Zbl 1061.11046; Zbl 1161.11371
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