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Positive lower density of large prime gaps
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed , throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where increases have positive lower density, answering Erdős and Prachar.
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Positive lower density of large prime gaps
Sep 25, 2026·main.pdf
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed , where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which increases have positive lower asymptotic density, answering a question of Erdős and Prachar.