The fourth and sixth power mean of the classical Kloosterman sums (Q616424)

Let \(S(m,n;q)\) be the classical Kloosterman sums and \(q\geq 3\). The paper obtains: For \((n,q)=1\), then \[ \sum\limits_{m=1}^{q}|S(m,n;q)|^{4}=3^{\omega(q)}q^{2}\varphi(q)\prod\limits_{p\|q}\left(\frac{2}{3}-\frac{1}{3p}-\frac{4}{3p(p-1)}\right), \] where \(\omega(q)\) denotes the number of all different prime divisors of \(q\) and \(\varphi(q)\) is the Euler function. The paper also obtains sixth power mean of the classical Kloosterman sums. The proof is elementary.