Under the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, a positive proportion of zeta zeros begin r consecutive gaps of size at least 2πc/log T, for any fixed r and small enough c.
Davenport, Multiplicative Number Theory , Graduate Texts in Math
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Consecutive moderate gaps between zeros of the Riemann zeta function
Under the Riemann Hypothesis and Montgomery's Pair Correlation Conjecture, a positive proportion of zeta zeros begin r consecutive gaps of size at least 2πc/log T, for any fixed r and small enough c.