On Heilbronn's exponential sum (Q2871307)

In the paper the author proves a new upper bound for Heilbronn's exponential sum and obtain some applications of his result to a distribution of Fermat quotients.NEWLINENEWLINEFor \(p\) a prime number, Heilbronn's exponential sum is defined as NEWLINENEWLINE\[ S(a) = \sum_{n=1}^p e^{2\pi i(an^p/p^2)}. \tag{1} \]NEWLINENEWLINEIn previous papers [\textit{D. R. Heath-Brown}, Prog. Math. 139, 451--463 (1996; Zbl 0857.11041), \textit{D. R. Heath-Brown} and \textit{S. Konyagin}, Q. J. Math. 51, No. 2, 221--235 (2000; Zbl 0983.11052)] a nontrivial upper bound for the sum \(S(a)\) was obtained.NEWLINENEWLINETheorem 1. Let \(p\) be a prime, and \(a\not\equiv 0\pmod p\). ThenNEWLINE\[ |S(a)| \ll p^{7/8}. \] NEWLINENEWLINEIgor Shparlinski asked the author about the possibility of an improvement of TheoremNEWLINE1. The main result of the present paper under the same conditions on \(p\) and \(a\) is NEWLINEthe estimate NEWLINENEWLINE\[ |S(a)| \ll p^{59/68} \log^{5/34} p.\]