Let $\Delta(x)$ denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be $\Delta^*(x)=-\Delta(x)+2\Delta(2x)-\frac12\Delta(4x)$. We show that \[ ıt_{T}^{T+H}\Delta^*\Big(\frac T{2i}\Big)\Big|\zeta\Big(\frac12+it\Big)\Big|^2dtl HT^{1/6}og^{7/2}T\qquad(T^{2/3}eq H=H(T)eq T), \] \[ ıt^T_0\Delta(f)\Big|\zeta\Big(\frac12+it\Big)\Big|^2dtl T^{9/8}(og T)^{5/2}, \] and obtain asymptotic formulae for \[ ıt^T_0\Big(\Delta^*\Big(\frac t{2i}\Big)\Big)^2\Big|\zeta\Big(\frac12+it\Big)\Big|^2dt,\qquadıt^T_0\Big(\Delta^*\Big(\frac t{2i}\Big)\Big)^3\Big|\zeta\Big(\frac12+it\Big)\Big|^2dt. \] The importance of the $\Delta^*$-function comes from the fact that it is the analogue of $E(T)$, the error term in the mean square formula for $\big|\zeta\big(\frac12+it\big)\big|^2$. We also show, if $E^*(T)=E(T)-2\pi\Delta^*(T/(2\pi))$, \[ ıt^T_0E^*(t)E^j(t)\Big|\zeta\Big(\frac12+it\Big)\Big|^2dtl_{j,ǎrepsilon}T^{7/6+j/4+ǎrepsilon}\qquad(j=1,2,3). \]