On an identity associated with Weil's estimate and its applications (Q1017391)

For integers \(q\geq 2\) and \(n\) put \[ G(n,\chi;q)=\sum_{a=1}^q \chi(a)e(na/q), \] where, as usual, \(e(z)=e^{2\pi i z}\) and \(\chi\) is a Dirichlet character modulo \(q\). The main results in the paper under review are two identities for \[ \sum_{\chi \bmod p} |G(n,\chi,q)|^{2l}\quad \text{for }l=3,4. \] These identities involve the sums \[ N=\sum_{a=2}^{p-1}\sum_{c=1}^{p-1} \Bigl({{a^2-c^2}\over {p}}\Bigr)\Bigl({{a^2-1}\over {p}}\Bigr)\Bigl({{c^2-1}\over {p}}\Bigr) \] and \[ T=\sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{d=1}^{p-1} \Bigl({{a^2-b^2}\over {p}}\Bigr)\Bigl({{a^2-d^2}\over {p}}\Bigr) \Bigl({{b^2-1}\over {p}}\Bigr)\Bigl({{d^2-1}\over {p}}\Bigr). \] As corollaries, the author deduces formulas for \[ \sum_{\chi \bmod p}\sum_{m=1}^p |C(m,n,\chi,q)|^{2l}\quad\text{for } l=3, 4, \] where \[ C(m,n,\chi,q)=\sum_{a=1}^q \chi(a)e\Bigl({{ma^k+na}\over {p}}\Bigr). \]