Titchmarsh's Method for the Approximate Functional Equations for zeta '(s)(2), zeta(s) zeta ''(s), and zeta '(s) zeta ''(s)

摘要

Let zeta(s) be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for zeta(2)(s) with error term O(x(1/2) (sigma)((x + y)/vertical bar t vertical bar(1/4) log vertical bar t vertical bar), where -1/2 < alpha < 3/2, x, y >= 1, xy = (vertical bar t vertical bar/2 pi)(2). Later, in 1938, Titchmarsh improved the error term by removing the factor ((x + y)/vertical bar t vertical bar(1/4). In 1999, Hall showed the approximate functional equations for zeta'(s)(2), zeta((S) zeta ''(S), and zeta'(s)zeta ''(s) (in the range 0 < sigma < 1) whose error terms contain the factor ((x + y)/vertical bar t vertical bar)(1/4). In this paper we remove this factor from these three error terms by using the method of Titchmarsh.