Large prime gaps and probabilistic models

Abstract

Abstract We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:math> . Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

Locations

• Inventiones mathematicae - View - PDF
• Inventiones mathematicae - View - PDF
• Inventiones mathematicae - View - PDF