Chowla, Sarvadaman On Kloosterman’s sum. (English) Zbl 0157.09001 Norske Vid. Selsk. Forhdl. 40, 70-72 (1967). The author shows that if \(\chi\) is any non-quadratic character modulo the odd prime \(p\) then the generalised Kloosterman sum satisfies the identity \[ \left|\sum_{n=1}^{p-1} \chi(m)e^{2\pi i(am-m^{-1})/p}\right| = \left|\sum_{n=0}^{p-1} e^{4\pi in}\chi(m) \bar\chi(n^2+a)\left(\frac{n^2+a}{p}\right)\right| \tag{1} \] and thus he estimates the Kloosterman sum by applying the well-known estimate \(2\sqrt{p}\) (due to A. Weil) to the right-hand side. (1) has been also proved by H. Davenport in [J. Reine Angew. Math. 169, 158–176 (1933; Zbl 0006.29501)]. Reviewer: David A. Burgess (Nottingham) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 29 Documents MSC: 11L05 Gauss and Kloosterman sums; generalizations 11L07 Estimates on exponential sums Keywords:Kloosterman sum Citations:Zbl 0006.29501 PDFBibTeX XMLCite \textit{S. Chowla}, Norske Vid. Selsk. Forhdl. 40, 70--72 (1967; Zbl 0157.09001)