Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals

2021/12/21 by Kaisa Matomäki, Matomäki, Kaisa, Jori Merikoski +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2112.11412

openalex publication_date 2021/12/21 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for ∑_{n ≤ X} Λ(n) Λ(± n+h) an asymptotic formula which holds uniformly for h = O(X). Such an asymptotic formula has been previously obtained only for fixed h in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large h we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

Cited by

• Primes in arithmetic progressions under the presence of Landau-Siegel zeroes
• The exceptional set of the Goldbach problem

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