On the general Kloosterman sum and its fourth power mean. (Q1421305)

Let \(q \geq 3\) be a positive integer and let \(\chi\) denote a Dirichlet character mod \(q\). For any integers \(m\) and \(n\), the general Kloosterman sum \(S(m ,n , \chi ; q)\) is defined as follows: \[ S(m,n, \chi ; q) = \sideset\and{^*}\to\sum_{a=1}^q \chi(a) e\left(\frac {ma+n \overline{a}}{q} \right), \] where \(e(y) = e^{2 \pi i y}\) and \(\sum^*\) denotes the summation over all \(a\) such that \((a,q) =1\), \(a \overline{a} \equiv 1\) mod \(q\). In the present paper the author derives an exact formula for the expression \[ \sum\limits_{\chi \bmod q} \sum_{m=1}^q | S(m,n, \chi ; q) | ^4 . \]