On the mean square of the error term in the approximate functional equation for \(\zeta^ 2(s)\) (Q1358288)

Let \(s={1\over 2}+ it\) with \(t>0\), and write \[ \zeta^2(s)= \sum_{n\leq t/2\pi} d(n)n^{-s}+ \chi^2(s) \sum_{n\leq t/2\pi} d(n)n^{s-1}+ R(t). \] It has been shown by \textit{I. Kiuchi} [Arch. Math. 67, 126-133 (1996; Zbl 0857.11043)] that \(\int^t_0|R(t)|^2dt= CT^{{1\over 2}}+K(T)\), with an explicit constant \(C\), and \(K(T)\ll\log^4T\). The present paper shows that \[ \int^T_0 R(t)dt= E_0T\log^3T+ E_1T\log^2T+ O(T\log T), \] with a negative constant \(E_0\). This result is inspired by the analogous estimate for the Dirichlet divisor problem, by \textit{Y.-K. Lau} and \textit{K.-M. Tsang} [J. Théor. Nombres Bordx. 7, 75-92 (1995; Zbl 0844.11059)].