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Prime Tuples and Siegel Zeros

Thomas Wright

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.

Prime Tuples and Siegel Zeros

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.
Paper Structure (15 sections, 10 theorems, 84 equations)

This paper contains 15 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Introduction: Siegel Zeroes
  3. Main Result
  4. Methods
  5. Remarks
  6. Outline
  7. Shifting the Primes
  8. The Maynard-Tao Sieve
  9. The Sum $S$
  10. The Sum $S_1$
  11. The Sum $S_2$
  12. Bounds for $H_m$
  13. Bounds for $H_m$ for $m=3,4,5$
  14. Bounds for $H_m$ for $m=2$
  15. Acknowledgements

Key Result

Theorem 1

(Landau 1918) There exists a constant $c$ such that for any $q$ and any real-valued character $\chi_q$ mod $q$, there exists at most one $s=\sigma+it$ for which $L(s,\chi_q)=0$ and .

Theorems & Definitions (12)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem 4.1
  • Corollary 4.2
  • Lemma 7.1
  • proof
  • Lemma 8.1
  • proof
  • ...and 2 more