The mean value formula for the approximate functional equation of \(\zeta^ 2(s)\) in the critical strip. II (Q1924957)

The author continues the investigations concerning the function \((s= \sigma + it,\;0\leq \sigma\leq 1\), \(t\geq 1\), \(x=t/(2\pi))\) \[ R(s;x): = \zeta^2(s)-\sum_{n\leq x}{'}d(n)n^{-s}-\chi^2 (s)\sum_{n\leq x}{'}d(n)n^{s-1}, \quad \chi(s) = \zeta(s)/ \zeta(s-1), \] begun in the first part of this work [Arch. Math. 64, 316-322 (1995; Zbl 0822.11059)]. Here as usual \(d(n)\) is the number of divisors of \(n\) and \(\sum'\) means that the last term in the sum is to be halved if \(x\) is an integer. If \(K(T)\) is defined, for a suitable constant \(E>0\), by \[ \int^T_1|R(1/2 + it;x) |^2 dt = ET^{1/2} + K(T), \] then the author [J. Number Theory 45, 312-319 (1993; Zbl 0787.11034)] and a little later the reviewer [Acta Arith. 65, 137-145 (1993; Zbl 0781.11036)] proved that \(K(T)=O(\log^5T)\). Now the author improves this bound to \(K(T) = O(\log^4 T)\). Since the reviewer (in a work to appear) proved that \[ \int^T_0 K(t)dt=E_0T \log^3T+E_1T \log^2 T+O (T\log T), \quad (E_0<0), \] the author's bound is close to being best possible. In fact the author obtains explicit expressions for \[ \int^T_1 \bigl|R(s;x) \bigr|^2dt\quad\bigl({\textstyle{1\over 2}} < \sigma \leq 1\bigr) \] involving the function \(K(T)\), which are hitherto the sharpest ones. To achieve this, use is made of \textit{Y. Motohashi}'s explicit formula for \(R(s;x)\) [Lectures on the Riemann-Siegel formula, Ulam Seminar, Colorado University, Boulder (1987)], which is fundamental in problems involving the function \(R(s;x)\).