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.

24 pages, 5 figures, 4 tables. Ver.3: to appear in Journal of Integer Sequences, vol.16 (2013)

Cited by in corpus (6)

• The distribution of maximal prime gaps in Cramer's probabilistic model of primes
• On the nth record gap between primes in an arithmetic progression
• On the distribution of maximal gaps between primes in residue classes
• On the first occurrences of gaps between primes in a residue class
• The distribution of prime numbers and tuples in the natural numbers
• Tables of record gaps between prime constellations