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.
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
\Bibitem{Pin12}
\by J\'anos~Pintz
\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
\serial Trudy Mat. Inst. Steklova
\yr 2012
\vol 276
\pages 227--232
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2012
\vol 276
\pages 222--227
\crossref{https://doi.org/10.1134/S008154381201018X}
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Citation in format AMSBIB
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