On the mean value of general Cochrane sum (Q963127)

The general Cochrane sum is defined by \(C(h,\chi_{1},p)=\sum_{a\leq p-1}\chi_{1}(a)((\frac{\bar{a}}{p}))((\frac{ah}{p}))\). The authors obtain \[ \sum_{h\leq p-1}C^{2}(h,\chi_{1},p)=\frac{1}{180}\prod_{p_{1}\in\mathcal{A}}\left(\frac{p_{1}^{2}+1}{p_{1}^{2}-1}\right)^{2} +O(p^{1+\varepsilon}), \] where \(\mathcal{A}\) is the set of quadratic residues of \(p\) and \(p_{1}\neq p\).