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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 276, Pages 227–232 (Mi tm3363)  

Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II

János Pintz

Alfréd Rényi Institute of Mathematics, Hungary Academy of Sciences, Budapest, Hungary
References:
Abstract: In an earlier work it was shown that the Elliott–Halberstam conjecture implies the existence of infinitely many gaps of size at most $16$ between consecutive primes. In the present work we show that assuming similar conditions not just for the primes but for functions involving both the primes and the Liouville function, we can assure not only the infinitude of twin primes but also the existence of arbitrarily long arithmetic progressions in the sequence of twin primes. An interesting new feature of the work is that the needed admissible distribution level for these functions is just $3/4$ in contrast to the Elliott–Halberstam conjecture.
Received in October 2011
English version:
Proceedings of the Steklov Institute of Mathematics, 2012, Volume 276, Pages 222–227
DOI: https://doi.org/10.1134/S008154381201018X
Bibliographic databases:
Document Type: Article
UDC: 511.337
Language: English
Citation: János Pintz, “Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II”, Number theory, algebra, and analysis, Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 276, MAIK Nauka/Interperiodica, Moscow, 2012, 227–232; Proc. Steklov Inst. Math., 276 (2012), 222–227
Citation in format AMSBIB
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\paper Are there arbitrarily long arithmetic progressions in the sequence of twin primes?~II
\inbook Number theory, algebra, and analysis
\bookinfo Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday
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\vol 276
\pages 227--232
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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