Let \[ \int^ T_ 0 \left | \zeta \Bigl( {1 \over 2} + it \Bigr) \right |^ 4dt = Tf (\log T) + E_ 2(T), \] where \(f\) is an appropriate quartic polynomial. It is shown here that \[ \int^ T_ 0 E_ 2(t)^ 2dt \ll T^ 2 (\log T)^ C \] for some constant \(C\). This remarkable result implies the estimates \(E_ 2 (T) \ll T^{2/3} (\log T)^ C\), and hence \(\zeta ({1 \over 2} + it) \ll t^{1/6} (\log t)^ C\), as well as the bound \[ \int^ T_ 0 \left | \zeta \Bigl( {1 \over 2} + it \Bigr) \right |^{12} dt \ll T^ 2 (\log T)^ C, \] with differing values of \(C\). Further theorems describe the mean value of the error terms for \(\sum^ N_{n = 1} d(n) d(n+k)\) and \(\sum^{N- 1}_{n=1} d(n)d(N-n)\). In particular, the latter has an asymptotic formula with an error term which is \(O(N^{{1 \over 2} + \varepsilon})\) in mean.
The proofs use the spectral theory of the non-Euclidean Laplacian.