Writing the approximate functional equation for \(\zeta(s)^2\) as \[ \zeta(s)^2=S(s,x)+\chi(s)^2 S(1-s,y)+R(s,x), \] where \(S(s,x)=\sum_{n\leq x}d(n)n^{-\sigma}\), \(xy=\tau^2\), \(\tau =t/2\pi\) and \(\chi(s)=2^s \pi^{s-1} \sin(\pi s/2) \Gamma(1-s),\) the author proves that, if \(0\leq \sigma \leq 1\), \(t\geq 10\) and \(1\leq x,y\leq \tau^2\), then \[ R(s,x) \ll \sqrt{t} x^{-s} \min (1,t^{-1} x)\ L \log (tx^{-1}+t^{-1}x)+t^{-1} x^{1-\sigma}\ (1+tx^{-1})\ \min (x^{\varepsilon}+L,y^{\varepsilon}+L), \] with \(L=\log t\). The proof is based on a smoothed version of Titchmarsh's method. The method of the present paper enables the author to obtain also the following result, due to \textit{Y. Motohashi} (to appear): for \(0\leq \sigma \leq 1\) and \(t\geq 10\) one has \[ \chi(1-s)R(s,x) = -\sqrt{2} t^{-1/2} \Delta (\tau)+O(t^{-1/4}) \] where \(\Delta (x)=\sum_{n\leq x}d(n)- x(\log x+2\gamma -1)\). An advantage of the present method is that it is based only on the functional equation for \(\zeta(s)^2\), and indeed similar results can be proved for Dirichlet series satisfying a functional equation of Riemann's type.