Abstract
Contrary to what would be predicted on the basis of Cramér’s model concerning the distribution of prime numbers, we develop evidence that the distribution of ψ(x+H)−ψ(x), for 0≤x≤N, is approximately normal with mean ∼H and variance ∼H log N/H, when Nδ≤H≤N1−δ.
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Communicated by P. Sarnak
Dedicated to Freeman Dyson, with best wishes on the occasion of his eightieth birthday
Research supported by NSF grants DMS–0070720 and DMS–0244660.
Research supported by the American Institute of Mathematics and NSF grants.
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Montgomery, H., Soundararajan, K. Primes in Short Intervals. Commun. Math. Phys. 252, 589–617 (2004). https://doi.org/10.1007/s00220-004-1222-4
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DOI: https://doi.org/10.1007/s00220-004-1222-4