A uniform estimate of the error term in the mean square of the Riemann zeta-function

Abstract

Let ((s), s = cr + it, denote the Riemann zeta-function, and let, for T ~> 2,The asymptotic formulas for I~ (T) as T --+ oo take an important place in the theory of the Riemann zeta-function. A large number of papers is devoted to the case cr = 1/2. Letwhere y denotes the Euler constant. It is known that E(T) = O(T ~)with ot ~< 1/3, and on the other hand it is proved [1] (see also [3]) that E(T) = ~'2(T1/4). The best known result in this direction asserts [2] that E(T) = O (T 7/22+s)for every s > 0.The case (r > 1/2 has been considered in [13]. LetIt has been obtained in [13] that, for 1/2 < cr < 3/4