Linear Equations in Primes
Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -&...
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| Format: | Journal article |
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2006
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| author | Green, B Tao, T |
| author_facet | Green, B Tao, T |
| author_sort | Green, B |
| collection | OXFORD |
| description | Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -> \infty, for the number of integer points n in K for which the integers \psi_1(n),...,\psi_t(n) are simultaneously prime. This implies many other well-known conjectures, such as the Hardy-Littlewood prime tuples conjecture, the twin prime conjecture, and the (weak) Goldbach conjecture. <p> In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms \psi_1,...,\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 ``non-degenerate'' configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the Inverse Gowers-norm conjecture GI(s) and the Mobius and Nilsequences Conjecture MN(s), where s \in {1,2,...} is the complexity of the system and measures the extent to which the forms \psi_i depend on each other. For s = 1 these are essentially classical, and the authors recently resolved the cases s = 2.<p> Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally for any (non-degenerate) problem involving two linear equations in four prime unknowns.</p></p> |
| first_indexed | 2024-03-07T03:55:03Z |
| format | Journal article |
| id | oxford-uuid:c296e0ff-026a-4123-a03c-c202d0889b24 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:55:03Z |
| publishDate | 2006 |
| record_format | dspace |
| spelling | oxford-uuid:c296e0ff-026a-4123-a03c-c202d0889b242022-03-27T06:10:06ZLinear Equations in PrimesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c296e0ff-026a-4123-a03c-c202d0889b24Symplectic Elements at Oxford2006Green, BTao, TConsider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -> \infty, for the number of integer points n in K for which the integers \psi_1(n),...,\psi_t(n) are simultaneously prime. This implies many other well-known conjectures, such as the Hardy-Littlewood prime tuples conjecture, the twin prime conjecture, and the (weak) Goldbach conjecture. <p> In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms \psi_1,...,\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 ``non-degenerate'' configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the Inverse Gowers-norm conjecture GI(s) and the Mobius and Nilsequences Conjecture MN(s), where s \in {1,2,...} is the complexity of the system and measures the extent to which the forms \psi_i depend on each other. For s = 1 these are essentially classical, and the authors recently resolved the cases s = 2.<p> Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally for any (non-degenerate) problem involving two linear equations in four prime unknowns.</p></p> |
| spellingShingle | Green, B Tao, T Linear Equations in Primes |
| title | Linear Equations in Primes |
| title_full | Linear Equations in Primes |
| title_fullStr | Linear Equations in Primes |
| title_full_unstemmed | Linear Equations in Primes |
| title_short | Linear Equations in Primes |
| title_sort | linear equations in primes |
| work_keys_str_mv | AT greenb linearequationsinprimes AT taot linearequationsinprimes |