On the mean-square of the Riemann zeta-function on the critical line

Abstract

The function E(T) is used to denote the error term in the mean-square estimate for the Riemann zeta-function on the half-line. In this paper we will prove a variety of new results concerning this function. The general aim is to extend the analogy of this function with the error term in Dirichlet's divisor problem. There are three main themes that we stress. The first theme is representations for the integral E1(T) = ∫0T E(t) dt. The forms these take are similar to (but more complicated than) the analogous formulas due to Voronoi in the divisor problem. The proof proceeds somewhat as the proof Atkinson used to get his representation for E(T) itself, and is about as difficult. The extra averaging does not seem to aid the method significantly, as it did for Voronoi. The second theme is upper and lower bound results. Essentially we show that the best bounds known in the divisor problem also hold for this function as well. These include both omega-plus and omega-minus results for E(T) and for E1(T). The results for E1(T) completely determine its order. The methods used here are again similar to ones used in the divisor problem. However, some recent innovations are needed to account for the lack of arithmetical structure and the complicated natures of our representation for E1(T) and Atkinson's for E(T). Finally, we prove a mean-square estimate for E1(T). This estimate indicates that this function frequently achieves its maximal order. © 1989.