Aleksandar Ivić

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 Dirichlets divisor problem. There are three main themes that we stress. The first theme is representations for the integral E 1 ( T ) = ∫ 0 T 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 E 1 ( T ). The results for E 1 ( 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 E 1 ( T ) and Atkinsons for E ( T ). Finally, we prove a mean-square estimate for E 1 ( T ). This estimate indicates that this function frequently achieves its maximal order.

The general divisor problem

Let Δk(x) = Δ(a1, …, ak; x) be the error term in the asymptotic formula for the summatory function of the general divisor function d(a1, …, ak; n), which is generated by ζ(a1s) … ζ(aks) (1 ≤ a1 ≤ … ≤ ak are fixed integers). Precise omega results for the mean square integral ∫1x Δk2(x) dx are given, with applications to some specific divisor problems.

On the riemann zeta-function and the divisor problem

AbstractLet Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of

Estimates of convolutions of certain number-theoretic error terms

\left| {\varsigma \left( {\tfrac{1}{2} + it} \right)} \right|

On a problem connected with zeros of ζ(s) on the critical line

. If

On some problems involving Hardy's function

E^* \left( t \right) = E\left( t \right) - 2\pi \Delta ^* \left( {t / 2\pi } \right)