A1 Refereed original research article in a scientific journal
The Hardy-Littlewood-Chowla conjecture in the presence of a Siegel zero
Authors: Tao Terence, Teräväinen Joni
Publisher: WILEY
Publication year: 2022
Journal: Journal of the London Mathematical Society
Journal name in source: JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
Journal acronym: J LOND MATH SOC
Number of pages: 62
ISSN: 0024-6107
eISSN: 1469-7750
DOI: https://doi.org/10.1112/jlms.12663
Web address : https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.12663
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/176032641
Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy-Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers.., and any distinct integers h(1), ... , h(k), h'(1), ... , h'(l), we establish an asymptotic formula forSigma(n <= x) Lambda(n + h(1)) ... Lambda(n + h(k))lambda(n + h'(1)) ... lambda(n + h'(l))for any 0 <= k <= 2 and l >= 0. Specializing to either l = 0 or.. = 0, we deduce the previously known results on the Hardy-Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to Heath-Brown and Chinis, respectively. The range of validity of our asymptotic formula is wider than in these previous results.
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