An Unconditional large gap between the Zeros of the Riemann Zeta-Function and Existence of Conditional Large Gaps

TL;DR

This paper improves the known unconditional lower bound for large gaps between zeros of the Riemann zeta-function and explores conditional bounds based on moment predictions, revealing that zeros can be significantly spaced apart.

Contribution

It establishes a new unconditional lower bound of 3.5555 for large gaps and derives explicit conditional bounds up to the 15th zero, using integral inequalities and moment hypotheses.

Findings

Unconditional large gap lower bound improved to 3.5555.

Conditional bounds for zeros gaps up to the 15th zero established.

Zeros can be at least 9.6435 times the average spacing apart when k=15.

Abstract

In this paper, we prove the lower bound of the unconditional large gap is 3.5555 which improves the obtained value 3.079 in the literature. Next, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted we establish a new explicit formula of the gaps and use it to establish some lower bounds for k=3,4,...,15. In particular it is proved that lower bound when k=15 is 9.6435 which means that the consecutive nontrivial zeros often differ by at least 9.6435 times the average spacing. The main results are proved by employed an integral inequality with a best constant proved by David Boyd.