Prime Tuples and Siegel Zeros

Under the assumption of infinitely many Siegel zeroes s with Re(s) > 1 − 1 (log q)r r for a sufficiently large value of r, we prove that there exist infinitely many m-tuples of primes that are ≪ e1.9828m apart. This ”improves” (in some sense) on the bounds of Maynard-Tao, Baker-Irving, and Polymath 8b, who found bounds of e3.815m unconditionally and me2m 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.