Maximal gaps between prime k-tuples: a statistical approach
Abstract
Combining the Hardy-Littlewood k-tuple conjecture with a heuristic application of extreme-value statistics, we propose a family of estimator formulas for predicting maximal gaps between prime k-tuples. Computations show that the estimator a(log(x/a)-b) satisfactorily predicts the maximal gaps below x, where a is the expected average gap between the same type of k-tuples, a=O(log^k x). Heuristics suggest that maximal gaps between prime k-tuples near x are approximately a*log(x/a), and thus have the order O(log^{k+1}x). The distribution of maximal gaps around the trend curve a*log(x/a) is close to the Gumbel distribution. We explore two implications of this model of gaps: record gaps between primes and Legendre-type conjectures for prime k-tuples.
Keywords
Cite
@article{arxiv.1301.2242,
title = {Maximal gaps between prime k-tuples: a statistical approach},
author = {Alexei Kourbatov},
journal= {arXiv preprint arXiv:1301.2242},
year = {2013}
}
Comments
24 pages, 5 figures, 4 tables. Ver.3: to appear in Journal of Integer Sequences, vol.16 (2013)