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

We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta function. More precisely, for every fixed $0<c\le 1\mathrm{/}4$, the lower asymptotic densities, normalized by $N(T)$, of occurrences of two, three, and four consecutive gaps, each of length at least $2\pi c\mathrm{/}\log T$, are at least $0.3450014\dots$, $0.1812517\dots$, and $0.0175021\dots$, respectively. Here zeros are ordered by ordinate and counted with multiplicity, and $N(T)$ denotes the number of nontrivial zeros with ordinates in $(0,T]$. This answers unconditionally and quantitatively a question studied by Gonek and Sahay under the Riemann hypothesis and the pair correlation conjecture. In particular, at the time of their work, it was not known unconditionally that even two consecutive moderate gaps occur infinitely often for any fixed $c>0$. The proof combines Lamzouri’s Hilbert-space formulation of the Montgomery–Taylor second-moment method with local spectral estimates for simple critical-line zeros and a cluster-counting argument that accounts for multiple zeros and zeros off the critical line.