Estimates for complete exponential sums of special types (Q2760485)

The paper is concerned with the exponential sums NEWLINE\[NEWLINE S_{p}(m,n)= \sum_{x=1}^{p} e^{2 \pi i(ax^{m}+bx^{n})/p} NEWLINE\]NEWLINE and with exponential sums of Heilbronn's type NEWLINE\[NEWLINE H_{p}(m,n)= \sum_{x=1}^{p} e^{2 \pi i (ax^{pm}+bx^{pn})/p^{2}}, NEWLINE\]NEWLINE where \(p\) is a prime, \(m,n,a,b\) are integers with \(m > n > 0\) and \(ab \not\equiv 0 \pmod p\). NEWLINENEWLINENEWLINEIf \( \delta=(m,p-1)\) and \( \tau=(n,p-1)\) the author proves that NEWLINE\[NEWLINE |S_{p}(m,n)|\leq p \cdot \min \left( \sqrt{( \delta, \tau) \tau/ \delta}, \sqrt{( \delta, \tau) \delta/ \tau} \right)+1. NEWLINE\]NEWLINE As a consequence he shows that if \(p-1 \equiv 0 \pmod m\), \(m > \sqrt{p}\) and \((m,n)=1\), then NEWLINE\[NEWLINE |S_{p}(m,n)|\leq p \cdot ( \tau/m)^{1/2}+1. NEWLINE\]NEWLINE This estimate is stronger than the corresponding bounds of \textit{L. J. Mordell} [Q. J. Math., Oxf. Ser. 3, 161-167 (1932; Zbl 0005.24603)] and \textit{N. M. Akulinichev} [Sov. Math., Dokl. 6, 480-482 (1965); translation from Dokl. Akad. Nauk SSSR 161, 743-745 (1965; Zbl 0127.02102)]. Moreover, making use of \textit{D. R. Heath-Brown} and \textit{S. Konyagin}'s result [Q. J. Math. 51, 221-235 (2000; Zbl 0983.11052)] on estimation of Gaussian sums the author proves that if \(p-1 \equiv 0 \pmod m\) and \((m,n)=1\), then NEWLINE\[NEWLINE S_{p}(m,n) \ll m^{-3/8}p^{9/8}.NEWLINE\]NEWLINE This estimate is non-trivial for \(m \gg p^{1/3}\) and is stronger than Weil's bound NEWLINE\[NEWLINE |S_{p}(m,n)|\leq mp^{1/2} NEWLINE\]NEWLINE for \(m \gg p^{5/11}\). It follows from the last two inequalities that NEWLINE\[NEWLINE S_{p}(m,n) \ll p^{21/22} NEWLINE\]NEWLINE uniformly in \(m\) and \(n\). NEWLINENEWLINENEWLINEFinally, making use the argument of Akulinichev (loc. cit.) the author shows that if \( \delta=(m,p-1)\) and \( \tau= (n,p-1)\), then NEWLINE\[NEWLINE H_{p}(m,n) \ll \min \left(p \sqrt{( \delta, \tau)/ \delta}+ \tau^{1/2} p^{15/16}, p \sqrt{( \delta, \tau)/ \tau}+ \delta^{1/2}p^{15/16} \right). NEWLINE\]NEWLINE This result gives a non-trivial upper bound for \(H_{p}(m,1)\) when \((m,p-1) \rightarrow \infty\) as \(p \rightarrow \infty\).