Unconditional and Conditional Large Gaps between the zeros of the Riemann Zeta-Function

Abstract

In this paper, first by employing inequalities derived from the Opial inequality due to David Boyd with best constant, we will establish new unconditional lower bounds for the gaps between the zeros of the Riemann zeta function. Second, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted, we establish some explicit formulae for the lower bounds of the gaps between the zeros and use them to establish some new conditional bounds. In particular it is proved that the consecutive nontrivial zeros often differ by at least 6.1392 (conditionally) times the average spacing. This value improves the value 4.71474396 that has been derived in the literature.