Number theory · Oct 6, 2026

Positive lower density of large prime gaps

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed Clog⁡pnC\log p_n, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where pn/np_n/n increases have positive lower density, answering Erdős and Prachar.

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