Unconditional Bounds for Two, Three and Four Consecutive Moderate Gaps between Zeros of the Riemann Zeta Function

Abstract

We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta function. More precisely, for every fixed \(0<c\leq 1/4\), the lower asymptotic densities, normalized by \(N(T)\), of the occurrences of two, three, and four consecutive gaps, each of length at least \(2πc/\log T\), are at least \(0.3450014\ldots\), \(0.1812517\ldots\), and \(0.0175021\ldots\), respectively. Here, zeros are ordered by ordinate and counted with multiplicity, and \(N(T)\) denotes the number of nontrivial zeros with ordinates in \((0,T]\). These bounds address the corresponding cases of a question of Gonek and Sahay. The proof combines the methods of Alpöge--Furman and Lamzouri with local spectral estimates and cluster counting.

arXiv comment21 pages. Revised abstract and introduction; added a remark on Lamzouri's independent work; clarified notation and exposition; added author email addresses and corresponding author information; corrected typos. Main results unchanged