An improved estimate of the fourth power mean of the general \(3\)-dimensional Kloosterman sum mod \(p\) (Q2238279)

Let \(p>3\) be a prime. This paper proved that \[ \sum_{m=1}^{p-1}\sum_{\chi\bmod p}\left| \sum_{a=1}^{p-1}\sum_{b=1}^{p-1}\sum_{c=1}^{p-1} \chi(abc)\cdot e\left(\frac{a+b+c+m\overline{abc}}{p}\right) \right|^4=2p^8+O\left(p^7\right), \] where \(\chi\bmod p\) denotes the summation over all Dirichlet characters modulo \(p\), \(e(y)=\hbox{e}^{2\pi iy}\) and \(\overline{x}\) is the multiplicative inverse of \(x\) modulo \(p\).