Some Improved Estimates in the Dirichlet Divisor Problem from Bourgain's Exponent Pair

Abstract

The thesis work is a survey of recent developments on the famous error terms in the Dirichlet divisor problem. We consider the power moments of the Riemann zeta-function in the critical strip and we managed to obtain some new bound estimates for power moments using a recently obtained exponent pair by Jean Bourgain. Thus, applying the slight improvements on bounds of the power moment estimates and the order of the zeta-function in the critical strip, we obtain new improved bounds for the order of the error term in the Dirichlet divisor problem.