The mean value formula for the approximate functional equation of \(\zeta^ 2(s)\) in the critical strip
Let \(s= \sigma+ it\), \(0\leq\sigma \leq 1\), \(t\geq 1\), \(x=t/ (2\pi)\), and let \[ R(s;x):= \zeta^ 2 (s)- \mathop{{\sum}'}_{n\leq x} d(n) n^{- s}- \chi^ 2 (s) \mathop {{\sum}'}_{n\leq x} d(n) n^{s-1} \] denote the error term in the approximate functional equation for \(\zeta^ 2 (s)\), where \(\zeta(s)\) is the Riemann zeta-function, \(d(n)\) is the number of divisors of \(n\), \(\chi(s)= \zeta(s)/ \zeta(s-1)\), and the dash in summation means that the last summand is to be halved if \(x\) is an integer. Power moments of \(| R({1 \over 2}+ it; x)|\) were investigated by the author [Publ. Inst. Math., Nouv. Sér. 52, 10-12 (1992; Zbl 0793.11021)] and the reviewer [Acta Arith. 65, 137-145 (1993; Zbl 0781.11036)]. The mean square of \(| R({1\over 2}+ it; x)|\) was treated by the author and \textit{K. Matsumoto} [Acta Arith. 61, 337-345 (1992; Zbl 0761.11035)], and the result was substantially sharpened by the author [J. Number Theory 45, 312-319 (1993; Zbl 0787.11034)]. Now the author supplements the mean square results by proving that \[ \int_ 1^ T | R(s;x) |^ 2 dt= \begin{cases} A_ \sigma T^{3/2- 2\sigma}+ O(T^{1- 2\sigma} \log^ 4 T), \quad &0\leq \sigma < 1/2,\\ A_{1/2} T^{1/2}+ O(\log^ 5 T), &\sigma=1/2,\\ A_ \sigma T^{3/2- 2\sigma}+ O(1), &1/2<\sigma< 3/4,\\ B\log T+ O(1), &\sigma=3/4,\\ O(1), &3/4< \sigma\leq 1, \end{cases} \] where \(A_ \sigma\) and \(B\) are precisely defined constants. \{Reviewer's remark: \textit{K. Matsumoto} [Arch. Math. 64, 323-332 (1995; Zbl 0822.11061 below)] refined the \(O\)-terms above when \(1/2< \sigma\leq 1\)\}.
approximate functional equation for \(\zeta^ 2 (s)\)
scientific article; zbMATH DE number 756435