Geodesic
Are there arbitrarily long arithmetic progressions in the sequence of twin primes?~II
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 227-232.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{TM_2012_276_a17,
     author = {J\'anos Pintz},
     title = {Are there arbitrarily long arithmetic progressions in the sequence of twin {primes?~II}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {227--232},
     publisher = {mathdoc},
     volume = {276},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_276_a17/}
}
TY  - JOUR
AU  - János Pintz
TI  - Are there arbitrarily long arithmetic progressions in the sequence of twin primes?~II
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2012
SP  - 227
EP  - 232
VL  - 276
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2012_276_a17/
LA  - en
ID  - TM_2012_276_a17
ER  - 
%0 Journal Article
%A János Pintz
%T Are there arbitrarily long arithmetic progressions in the sequence of twin primes?~II
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2012
%P 227-232
%V 276
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2012_276_a17/
%G en
%F TM_2012_276_a17
János Pintz. Are there arbitrarily long arithmetic progressions in the sequence of twin primes?~II. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 227-232. http://geodesic.mathdoc.fr/item/TM_2012_276_a17/

[1] Chowla S., The Riemann hypothesis and Hilbert's tenth problem, Gordon and Breach, New York, 1965 | MR

[2] Elliott P.D.T.A., Halberstam H., “A conjecture in prime number theory”, Symposia mathematica. V. 4: Convegni del Dicembre del 1968 e del Marzo del 1969, Acad. Press, London; Ist. Naz. Alta Mat., Rome, 1970, 59–72 | MR

[3] Goldston D.A., Pintz J., Yıldırım C.Y., “Primes in tuples. I”, Ann. Math. Ser. 2, 170:2 (2009), 819–862 | DOI | MR | Zbl

[4] Green B., Tao T., “The primes contain arbitrarily long arithmetic progressions”, Ann. Math. Ser. 2, 167:2 (2008), 481–547 | DOI | MR | Zbl

[5] Hildebrand A.J., “Erdős' problems on consecutive integers”, Paul Erdős and his mathematics I, Bolyai Soc. Math. Stud., 11, J. Bolyai Math. Soc., Budapest, 2002, 305–317 | MR | Zbl

[6] Iwaniec H., “Prime numbers and $L$-functions”, Proc. Intern. Congr. Math., Madrid, 2006, V. 1, Eur. Math. Soc., Zürich, 2007, 279–306 | DOI | MR | Zbl

[7] Pintz J., “Are there arbitrarily long arithmetic progressions in the sequence of twin primes?”, An irregular mind: Szemerédi is 70, Bolyai Soc. Math. Stud., 21, eds. Ed. by I. Bárány, J. Solymosi, Springer, Berlin, 2010, 525–559 | DOI | MR | Zbl

[8] Pintz J., An approximation to the twin prime conjecture and the parity phenomenon, E-print, 2010, arXiv: 1004.1065v1 [math.NT] | MR

[9] Vaughan R.C., “An elementary method in prime number theory”, Recent progress in analytic number theory, Durham, 1979, V. 1, Acad. Press, London, 1981, 341–348 | MR

[10] Zhou B., “The Chen primes contain arbitrarily long arithmetic progressions”, Acta arith., 138:4 (2009), 301–315 | DOI | MR | Zbl