Reimann Hypothesis Forum

Prime Gaps, Cramér's Conjecture, and Elementary Sieve Bounds

Under RH, von Koch's bound gives p_{n+1} - p_n = O(√p_n log p_n), which is still far from Cramér's O((log p_n)^2) heuristic. Are there any readable expositions bridging the random model to Granville's refinement?

Granville's 1995 paper on Cramér's model is surprisingly accessible. The key insight is that divisibility by small primes introduces local correlations that bump the constant above 1, suggesting limsup (p_{n+1}-p_n)/(\log p_n)^2 ≥ 2e^{-\gamma} ≈ 1.1229.