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 An identity on the $2m$-th power mean value of the generalized Gauss sums Bull. Korean Math. Soc. 2012 Vol. 49, No. 6, 1327-1334 https://doi.org/10.4134/BKMS.2012.49.6.1327Published online November 30, 2012 Feng Liu and Quan-Hui Yang Nanjing Normal University, Nanjing Normal University Abstract : In this paper, using analytic method and the properties of the Legendre's symbol, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang [On the $2k$-th power mean value of the generalized quadratic Gauss sums, Bull. Korean Math. Soc. $\bf 48$ (2011), no. 1, 9--15]. Keywords : $2m$-th power mean, exact calculating formula, generalized quadratic Gauss sums MSC numbers : Primary 11L05; Secondary 11M20 Downloads: Full-text PDF