A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Siegel Zeros, Twin Primes, Goldbach's Conjecture, and Primes in Short Intervals
Tekijät: Matomäki Kaisa, Merikoski Jori
Kustantaja: OXFORD UNIV PRESS
Julkaisuvuosi: 2023
Journal: International Mathematics Research Notices
Tietokannassa oleva lehden nimi: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
Lehden akronyymi: INT MATH RES NOTICES
Artikkelin numero: rnad069
Sivujen määrä: 48
ISSN: 1073-7928
eISSN: 1687-0247
DOI: https://doi.org/10.1093/imrn/rnad069
Verkko-osoite: https://doi.org/10.1093/imrn/rnad069
Rinnakkaistallenteen osoite: https://research.utu.fi/converis/portal/detail/Publication/179777742
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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