Abstract
Consider a system $\Psi$ of nonconstant affine-linear forms $\psi_1,\dots,\psi_t: \mathbb{Z}^d \to \mathbb{Z}$, no two of which are linearly dependent. Let $N$ be a large integer, and let $K \subseteq [-N,N]^d$ be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as $N \to \infty$, for the number of integer points $ n \in \mathbb{Z}^d \cap K$ for which the integers $\psi_1( n),\dots,\psi_t( n)$ are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime.
In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms $\psi_1,\dots,\psi_t$ are affinely related; this excludes the important “binary” cases such as the twin prime or Goldbach conjectures, but does allow one to count “nondegenerate” configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (${\rm GI}(s)$) and the Möbius and nilsequences conjecture ($\operatorname{MN}(s)$), where $s \in \{1,2,\dots\}$ is the complexity of the system and measures the extent to which the forms $\psi_i$ depend on each other. The case $s=0$ is somewhat degenerate, and follows from the prime number theorem in APs.
Roughly speaking, the inverse Gowers-norm conjecture $\operatorname{GI}(s)$ asserts the Gowers $U^{s+1}$-norm of a function $f : [N] \rightarrow [-1,1]$ is large if and only if $f$ correlates with an $s$-step nilsequence, while the Möbius and nilsequences conjecture ${\rm MN}(s)$ asserts that the Möbius function $\mu$ is strongly asymptotically orthogonal to $s$-step nilsequences of a fixed complexity. These conjectures have long been known to be true for $s=1$ (essentially by work of Hardy-Littlewood and Vinogradov), and were established for $s=2$ in two papers of the authors. Thus our results in the case of complexity $s \leq 2$ are unconditional.
In particular we can obtain the expected asymptotics for the number of $4$-term progressions $p_1 < p_2 < p_3 < p_4 \leq N$ of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.
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[iwaniec-kowalski] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004.
@book {iwaniec-kowalski, MRKEY = {2061214},
AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
TITLE = {Analytic Number Theory},
SERIES = {Amer. Math. Soc. Colloq. Publ.},
NUMBER = {53},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {xii+615},
ISBN = {0-8218-3633-1},
MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
MRNUMBER = {2005h:11005},
MRREVIEWER = {K. Soundararajan},
ZBLNUMBER = {1059.11001},
} -
[kra-icm-lecture] B. Kra, "From combinatorics to ergodic theory and back again," in International Congress of Mathematicians. Vol. III, Eur. Math. Soc., Zürich, 2006, pp. 57-76.
@incollection {kra-icm-lecture, MRKEY = {2275670},
AUTHOR = {Kra, Bryna},
TITLE = {From combinatorics to ergodic theory and back again},
BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {\rm III}},
PAGES = {57--76},
PUBLISHER = {Eur. Math. Soc., Zürich},
YEAR = {2006},
MRCLASS = {37A30 (11B25 37A45)},
MRNUMBER = {2007m:37014},
MRREVIEWER = {Ilya D. Shkredov},
ZBLNUMBER = {1118.37010},
} -
[sasha-personal] A. Leibman, Personal communication.
@misc{sasha-personal,
author={Leibman, A.},
TITLE={Personal communication},
} -
[malcev] A. I. Mal’cev, "On a class of homogeneous spaces," Izvestiya Akad. Nauk. SSSR. Ser. Mat., vol. 13, pp. 9-32, 1949.
@article {malcev, MRKEY = {0028842},
AUTHOR = {Mal'cev, A. I.},
TITLE = {On a class of homogeneous spaces},
JOURNAL = {Izvestiya Akad. Nauk. SSSR. Ser. Mat.},
FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
VOLUME = {13},
YEAR = {1949},
PAGES = {9--32},
ISSN = {0373-2436},
MRCLASS = {20.0X},
MRNUMBER = {10,507d},
MRREVIEWER = {I. Kaplansky},
} -
[steiner] J. Steiner, "Über paralelle Flächen," Jbr. Preuss. Akad. Wiss., pp. 114-118, 1840.
@article{steiner,
author={Steiner, J.},
TITLE={Über paralelle Flächen},
JOURNAL={Jbr. Preuss. Akad. Wiss.},
YEAR={1840},
PAGES={114--118},
NOTE={(Ges. Werke Vol. II, Reiner, Berlin (1882), pp.~173--176},
} -
[szemeredi] E. Szemerédi, "On sets of integers containing no $k$ elements in arithmetic progression," Acta Arith., vol. 27, pp. 199-245, 1975.
@article {szemeredi, MRKEY = {0369312},
AUTHOR = {Szemer{é}di, E.},
TITLE = {On sets of integers containing no {$k$} elements in arithmetic progression},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {27},
YEAR = {1975},
PAGES = {199--245},
ISSN = {0065-1036},
MRCLASS = {10L10},
MRNUMBER = {51 \#5547},
MRREVIEWER = {S. L. G. Choi},
ZBLNUMBER = {0303.10056},
} -
[tao:ergodic]
T. Tao, "A quantitative ergodic theory proof of Szemerédi’s theorem," Electron. J. Combin., vol. 13, iss. 1, p. 99, 2006.
@article {tao:ergodic, MRKEY = {2274314},
AUTHOR = {Tao, Terence},
TITLE = {A quantitative ergodic theory proof of {S}zemerédi's theorem},
JOURNAL = {Electron. J. Combin.},
FJOURNAL = {Electronic Journal of Combinatorics},
VOLUME = {13},
YEAR = {2006},
NUMBER = {1},
PAGES = {Research Paper 99, 49 pp.},
ISSN = {1077-8926},
MRCLASS = {37A45 (11B25)},
MRNUMBER = {2007i:37016},
MRREVIEWER = {Bryna Kra},
URL = {http://www.combinatorics.org/Volume_13/Abstracts/v13i1r99.html},
} -
[tao-elescorial] T. Tao, "Arithmetic progressions and the primes." , 2006, vol. 2006 , Extra Vol., pp. 37-88.
@incollection{tao-elescorial,
author = {Tao, Terence},
TITLE = {Arithmetic progressions and the primes},
SERIES={Collect. Math.},
VOLUME={2006{\rm , Extra Vol.}},
YEAR={2006},
PAGES={37--88},
NOTE={Proc. 7th International Conference on Harmonic Analysis and Partial Differential Equations},
MRNUMBER={2007k:11151},
} -
[tao-coates] T. Tao, "Obstructions to uniformity and arithmetic patterns in the primes," Pure Appl. Math. Q., vol. 2, iss. 2, part 2, pp. 395-433, 2006.
@article {tao-coates, MRKEY = {2251475},
AUTHOR = {Tao, Terence},
TITLE = {Obstructions to uniformity and arithmetic patterns in the primes},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {2},
YEAR = {2006},
NUMBER = {2, part 2},
PAGES = {395--433},
ISSN = {1558-8599},
MRCLASS = {11N13 (11B25 37A45)},
MRNUMBER = {2007d:11106},
MRREVIEWER = {Ben Joseph Green},
ZBLNUMBER = {1105.11032},
} -
@misc{tao-gy-notes,
author = {Tao, Terence},
TITLE = {A remark on Goldston-Yıldırım correlation estimates},
URL={http://www.math.ucla.edu/~tao/preprints/Expository/gy-corr.dvi},
} -
[tao-hyper]
T. Tao, "A variant of the hypergraph removal lemma," J. Combin. Theory Ser. A, vol. 113, iss. 7, pp. 1257-1280, 2006.
@article {tao-hyper, MRKEY = {2259060},
AUTHOR = {Tao, Terence},
TITLE = {A variant of the hypergraph removal lemma},
JOURNAL = {J. Combin. Theory Ser. A},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {113},
YEAR = {2006},
NUMBER = {7},
PAGES = {1257--1280},
ISSN = {0097-3165},
CODEN = {JCBTA7},
MRCLASS = {05C35 (05C65 05C75 11B75 37A45)},
MRNUMBER = {2007k:05098},
MRREVIEWER = {Jozef Skokan},
DOI = {10.1016/j.jcta.2005.11.006},
ZBLNUMBER = {1105.05052},
} -
[tao-multiprime] T. Tao, "The Gaussian primes contain arbitrarily shaped constellations," J. Anal. Math., vol. 99, pp. 109-176, 2006.
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TITLE = {The {G}aussian primes contain arbitrarily shaped constellations},
JOURNAL = {J. Anal. Math.},
FJOURNAL = {Journal d'Analyse Mathématique},
VOLUME = {99},
YEAR = {2006},
PAGES = {109--176},
ISSN = {0021-7670},
CODEN = {JOAMAV},
MRCLASS = {11B25 (11B75 37A45)},
MRNUMBER = {2007j:11014},
MRREVIEWER = {Ben Joseph Green},
ZBLNUMBER = {1160.11006},
} -
[tao-vu] T. Tao and V. Vu, Additive Combinatorics, Cambridge: Cambridge Univ. Press, 2006.
@book {tao-vu, MRKEY = {2289012},
AUTHOR = {Tao, Terence and Vu, Van},
TITLE = {Additive Combinatorics},
SERIES = {Cambridge Stud. Adv. Math.},
NUMBER = {105},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2006},
PAGES = {xviii+512},
ISBN = {978-0-521-85386-6; 0-521-85386-9},
MRCLASS = {11-02 (05-02 05D10 11B13 11P70 11P82 28D05 37A45)},
MRNUMBER = {2008a:11002},
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ZBLNUMBER = {1127.11002},
} -
[van-der-corput]
J. G. van der Corput, "Über Summen von Primzahlen und Primzahlquadraten," Math. Ann., vol. 116, iss. 1, pp. 1-50, 1939.
@article {van-der-corput, MRKEY = {1513216},
AUTHOR = {van der Corput, J. G.},
TITLE = {Über {S}ummen von {P}rimzahlen und {P}rimzahlquadraten},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {116},
YEAR = {1939},
NUMBER = {1},
PAGES = {1--50},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {Contributed Item},
MRNUMBER = {1513216},
DOI = {10.1007/BF01597346},
ZBLNUMBER = {0019.19602},
} -
[vinogradov-1] I. M. Vinogradov, "Some theorems concerning the primes," , vol. 2, pp. 179-195, 1937.
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author={Vinogradov, I. M.},
TITLE={Some theorems concerning the primes},
VOLUME={2},
YEAR={1937},
PAGES={179--195},
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[zygmund] A. Zygmund, Trigonometric Series. Vol. I, II, Third ed., Cambridge: Cambridge Univ. Press, 2002.
@book {zygmund, MRKEY = {1963498},
AUTHOR = {Zygmund, A.},
TITLE = {Trigonometric Series. {V}ol. {\rm I},
{\rm II}},
SERIES = {Cambridge Mathematical Library},
EDITION = {Third},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2002},
PAGES = {xii; Vol. I: xiv+383 pp.; Vol. II: viii+364},
ISBN = {0-521-89053-5},
MRCLASS = {01A75 (42-02)},
MRNUMBER = {2004h:01041},
ZBLNUMBER = {0367.42001},
ZBLNUMBER = {0131.06703},
ZBLNUMBER = {0085.05601},
}