On the error term in the mean square formula for the Riemann zeta-function in the critical strip (Q1912216)

Let \(\zeta (s)\) be the Riemann zeta-function, \(1/2< \sigma= \text{Re} (s)< 1\), and for \(T\geq 2\) \[ E_\sigma (T)= \int^T_0 |\zeta (\sigma+ it)|^2 dt- \zeta (2\sigma) T- (2\pi)^{2\sigma -1} {{\zeta (2- 2\sigma)} \over {2- 2\sigma}} T^{2- 2\sigma}. \] The function \(E_\sigma (T)\), which represents the error term in the mean square formula for \(\zeta (s)\) in the critical strip, was introduced by the second author [Jap. J. Math., New Ser. 15, 1-13 (1989; Zbl 0684.10035)]. Subsequent study of this function includes the work of the first author [Lectures on mean values of the Riemann zeta-function, Mathematics. Tata Institute of Fundamental Research 82, Springer (1991; Zbl 0758.11036)]\ and of the second author and \textit{T. Meurman} [Acta. Arith. 64, 357-382 (1993; Zbl 0788.11035) and ibid. 68, 369-382 (1994; Zbl 0812.11049)]. In the present paper the authors prove first various upper bounds for \(E_\sigma (T),\) such as \[ E_\sigma (T)\ll T^{2(1- \sigma)/3} (\log T)^{2/9}) \quad (1/2 < \sigma< 1), \] which improve the corresponding bound in the aforementioned work of the first author. The second result is \[ E_\sigma (T)= \Omega_- (T^{3/ 4-\sigma} \exp (C(\log \log T)^{\sigma- 1/4} (\log \log \log T)^{\sigma- 5/4})) \qquad (C>0) \] for \(1/2< \sigma< 3/4\). This corresponds to the sharpest known \(\Omega_-\)-result in the classical Dirichlet divisor problem, due to \textit{K. Corrádi} and \textit{I. Kátai} [Magyar Tud. Akad., Mat. Fiz. Tud. Oszt. Közl. 17, 89-97 (1967; Zbl 0163.04103)]. Both results proved in the present paper are hitherto the sharpest ones, and together with the results mentioned above, they draw a fairly accurate picture of the behaviour of the fundamental function \(E_\sigma (T)\).