##
**
Maximal Gaps Between Prime ***k*-Tuples: A Statistical Approach

###
Alexei Kourbatov

JavaScripter.net/math

15127 NE 24th St #578

Redmond, WA 98052

USA

**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. 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 ±2*a*, where
*a* = *C*_{k} log^{k}*x*
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*(log^{k+1}*x*).
The distribution of maximal gaps around the Â“trendÂ” 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,
dvi,
ps,
latex
**

(Concerned with sequences
A005250
A091592
A113274
A113404
A192870
A200503
A201051
A201062
A201073
A201251
A201596
A201598
A202281
A202361.)

Received January 22 2013;
revised version received May 1 2013.
Published in *Journal of Integer Sequences*, May 9 2013.

Return to
**Journal of Integer Sequences home page**