In this important paper the author establishes an explicit formula for \[ I(T,\Delta)= (\Delta\sqrt{\pi})^{-1} \int_{-\infty}^ \infty |\zeta ({\textstyle {1\over 2}}+iT+it)|^ 4 e^{-(t/\Delta)^ 2} dt \qquad (00\), \[ I(T,\Delta)= \pi(2T)^{-1/2} \sum_{j=1}^ \infty \alpha_ j H_ j^ 3 ({\textstyle {1\over 2}}) \kappa_ j^{-1/2} \sin \left(\kappa_ j \log {{\kappa_ j} \over {4eT}} \right) \exp \left(-\left( {{\Delta\kappa_ j} \over {2T}}\right)^ 2 \right)+ O(\log^ B T) \] in the standard notation of spectral theory: \(H_ j(s)\) is the Hecke \(L\)-series, \(\{\kappa_ j^ 2+{1\over 4}\}\) is the discrete spectrum of the non-Euclidean Laplacian and \(\alpha_ j= |\rho_ j(1)|^ 2 \text{ ch}(\pi\kappa_ j)^{-1}\). However, the importance of the author's result is best reflected in the fact that it can be used to derive results on \(E_ 2(T)\), the error term in the asymptotic formula for \(\int_ 0^ T |\zeta({1\over 2}+it)|^ 4 dt\). Using the formula for \(I(T,\Delta)\) and a result of the author on spectral mean values [J. Number Theory 42, 258-284 (1992; Zbl 0759.11026)] the author and the reviewer proved [Proc. Japan Acad., Ser. A 66, 150-152 (1990; Zbl 0688.10037) and ``On the fourth power moment of the Riemann zeta-function'', J. Number Theory (in press)] that \(E_ 2(T)= \Omega(T^{1/2})\) (recently the author sharpened this to \(E_ 2(T)= \Omega_ \pm (T^{1/2}))\) and \(E_ 2(T)\ll T^{2/3} \log^ C T\). In another joint paper [``The mean square of the error term for the fourth power moment of the zeta-function'', Proc. Lond. Math. Soc. (in press)] it is proved that \[ \int_ 0^ T E_ 2^ 2(t)dt \ll T^ 2 \log^ C T. \] Thus although \(I(T,\Delta)\) is a weighted integral, and not directly an integral of \(|\zeta({1\over 2}+it)|^ 4\), the author's formula for \(I(T,\Delta)\) is of such strength that it permits one to extract from it quite precise results on \(E_ 2(T)\) and other related functions.