A1 Refereed original research article in a scientific journal
Siegel Zeros, Twin Primes, Goldbach's Conjecture, and Primes in Short Intervals
Authors: Matomäki Kaisa, Merikoski Jori
Publisher: OXFORD UNIV PRESS
Publication year: 2023
Journal: International Mathematics Research Notices
Journal name in source: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
Journal acronym: INT MATH RES NOTICES
Article number: rnad069
Number of pages: 48
ISSN: 1073-7928
eISSN: 1687-0247
DOI: https://doi.org/10.1093/imrn/rnad069(external)
Web address : https://doi.org/10.1093/imrn/rnad069(external)
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/179777742(external)
We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular, we prove forSigma(n <= X) Lambda(n)Lambda(+/- n+h)an asymptotic formula that 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 Teravainen (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.
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