Mean value results for the non-symmetric form of the approximate functional equation of the Riemann zeta-function (Q1335286)

The author proves the asymptotic formula \[ \int^ T_ 2 \left | R^* \left( {1 \over 2} + it; {\ell t \over 2 \pi k} \right) \right |^ 2dt = C_{k,\ell} T^{1/2} + Q \bigl( (k \ell)^{3/4} T^{1/4} \log^ 3T \bigr) \tag{1} \] with explicitly given constant \(C_{k,\ell} (>0)\), valid for integers \(1 \leq \ell \leq k\), \((k,\ell) = 1\), \(k \ell \leq T \log^{-20}T\), \(T\geq 2\). Here for \(s = \sigma + it\) \[ R^* \left( s;{\ell t \over 2 \pi k} \right) = \zeta^ 2 (s) - \sum_{n \leq \ell t/(2 \pi k)} d(n)n^{-s} - \chi^ 2 (s) \sum_{n \leq kt/(2 \pi \ell)} d(n)n^{s-1} \] denotes the error term in the non- symmetric form of the approximate functional equation for \(\zeta^ 2(s)\). An upper bound for the integral in (1), only without absolute value signs, is also derived.