Prime Tuples and Siegel Zeros

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 $s$ with $Re(s)>1-\frac{1}{(\log q)^{R}}$ for a sufficiently large value of $R$, we prove that there exist infinitely many $m$-tuples of primes that are $\ll e^{1.9828m}$ apart. This "improves" (in some sense) on the bounds of Maynard-Tao, Baker-Irving, and Polymath 8b, who found bounds of $e^{3.815m}$ unconditionally and $me^{2m}$ 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.