Maximal Gaps Between Prime k-Tuples: A Statistical Approach
15127 NE 24th St #578
Redmond, WA 98052
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. Extensive computations show that the estimator
a log(x/a) − ba satisfactorily predicts the maximal gaps below x, in most cases within
an error of ±2a, where
a = Ck logkx
is the expected average gap between the same type of k-tuples.
Heuristics suggest that maximal gaps between prime k-tuples near
x are asymptotically equal to a log(x/a),
and thus have the order O(logk+1x).
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.
Full version: pdf,
(Concerned with sequences
Received January 22 2013;
revised version received May 1 2013.
Published in Journal of Integer Sequences, May 9 2013.
Journal of Integer Sequences home page