A1 Refereed original research article in a scientific journal

Siegel Zeros, Twin Primes, Goldbach's Conjecture, and Primes in Short Intervals




AuthorsMatomäki Kaisa, Merikoski Jori

PublisherOXFORD UNIV PRESS

Publication year2023

JournalInternational Mathematics Research Notices

Journal name in sourceINTERNATIONAL MATHEMATICS RESEARCH NOTICES

Journal acronymINT MATH RES NOTICES

Article number rnad069

Number of pages48

ISSN1073-7928

eISSN1687-0247

DOIhttps://doi.org/10.1093/imrn/rnad069(external)

Web address https://doi.org/10.1093/imrn/rnad069(external)

Self-archived copy’s web addresshttps://research.utu.fi/converis/portal/detail/Publication/179777742(external)


Abstract
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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Last updated on 2024-26-11 at 23:31