
TL;DR
This paper demonstrates that assuming infinitely many Siegel zeroes leads to the existence of infinitely many prime m-tuples with bounded gaps, improving previous bounds and extending classical results on prime constellations.
Contribution
It establishes new bounds on prime gaps under the assumption of infinitely many Siegel zeroes, generalizing Heath-Brown's twin prime result and improving previous bounds from Maynard-Tao and others.
Findings
Existence of infinitely many prime m-tuples with gaps ^{1.9828m} under Siegel zero assumption
Improved upper bounds for gaps between prime triples, quadruples, quintuples, and sextuples
Extension of Heath-Brown's result linking Siegel zeroes to infinitely many twin primes
Abstract
Under the assumption of infinitely many Siegel zeroes with for a sufficiently large value of , we prove that there exist infinitely many -tuples of primes that are apart. This "improves" (in some sense) on the bounds of Maynard-Tao, Baker-Irving, and Polymath 8b, who found bounds of unconditionally and assuming the Elliott-Halberstam conjecture; it also generalizes a 1983 result of Heath-Brown that states that infinitely many Siegel zeroes would imply infinitely many twin primes. Under this assumption of Siegel zeroes, we also improve the upper bounds for the gaps between prime triples, quadruples, quintuples, and sextuples beyond the bounds found via Elliott-Halberstam.
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Taxonomy
TopicsAnalytic Number Theory Research · Mathematical Dynamics and Fractals · Algebraic Geometry and Number Theory
