On the mean values of certain character sums (Q2015368)

Let \(q\geq 5\) be an odd number, this paper gives asymptotic formulae to the character sums: \[ \mathop{\sum\nolimits^{*}}_{\substack{\chi \mod q,\\ \chi(-1)=-1}}\left|\sum_{1\leq a\leq q/4}a\chi(a)\right|^{4} \] and \[ \mathop{\sum\nolimits^{*}}_{\substack{\chi \mod q,\\ \chi(-1)=1}} \left|\sum_{1\leq a\leq q/4}a\chi(a)\right|^{4}, \] where \(\sum^{*}\) denotes the summation over primitive characters modulo \(q\). The authors express the character sum in terms of Gauss sums and \(L\)-functions, then the mean values of \(L\)-functions give the asymptotic formulae.