OP-00918 Open

The Hardy–Littlewood prime k-tuples conjecture

Let $b_1, \dots, b_m$ be $m$ distinct integers, and let $P(x; b_1, \dots, b_m)$ be the number of groups $n + b_1, \dots, n + b_m$ between $1$ and $x$ that consist wholly of primes. For a prime $q$, let $\nu(q)$ be the number of distinct residues of $b_1, \dots, b_m$ modulo $q$, and let $$ \mathfrak G(b_1, \dots, b_m) = \prod_{q \ge 2} \left(\frac{q}{q-1}\right)^{m-1} \frac{q - \nu(q)}{q-1}, \qquad \operatorname{Li}_m(x) = \int_2^x \frac{du}{(\log u)^m}, $$ the product running over the primes. Hardy and Littlewood deduce the formula below from an unproved hypothesis of theirs (Hypothesis X) on the behaviour, as $r \to 1$, of an associated power series in $r$.

Problem. Prove that if $\mathfrak G(b_1, \dots, b_m) \ne 0$, then $P(x; b_1, \dots, b_m) \sim \mathfrak G(b_1, \dots, b_m) \operatorname{Li}_m(x)$ as $x \to \infty$.

Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Theorem X1, derived there from their Hypothesis X.

Imported from doi.org/10.1007/BF02403921.

Classification

Primary: NT — Number Theory
Keywords: Hardy–Littlewood conjectures, prime constellations, prime k-tuples, singular series
Literature checked: Oct. 25, 2026 by clemens-admin

References

@article{hardy1923,
  author = {Hardy, G. H. and Littlewood, J. E.},
  title = {Some problems of `{P}artitio numerorum'; {III}: {O}n the expression of a number as a sum of primes},
  journal = {Acta Mathematica},
  volume = {44},
  pages = {1--70},
  year = {1923},
  doi = {10.1007/BF02403921}
}

@article{dickson1904,
  author = {Dickson, Leonard Eugene},
  title = {A new extension of {D}irichlet's theorem on prime numbers},
  journal = {Messenger of Mathematics},
  volume = {33},
  pages = {155--161},
  year = {1904}
}

@article{bateman1962,
  author = {Bateman, Paul T. and Horn, Roger A.},
  title = {A heuristic asymptotic formula concerning the distribution of prime numbers},
  journal = {Mathematics of Computation},
  volume = {16},
  pages = {363--367},
  year = {1962}
}

@article{gallagher1976,
  author = {Gallagher, P. X.},
  title = {On the distribution of primes in short intervals},
  journal = {Mathematika},
  volume = {23},
  pages = {4--9},
  year = {1976}
}

@article{hensley1974,
  author = {Hensley, Douglas and Richards, Ian},
  title = {Primes in intervals},
  journal = {Acta Arithmetica},
  volume = {25},
  pages = {375--391},
  year = {1973/74}
}

@article{zhang2014,
  author = {Zhang, Yitang},
  title = {Bounded gaps between primes},
  journal = {Annals of Mathematics},
  volume = {179},
  pages = {1121--1174},
  year = {2014}
}

@article{maynard2015,
  author = {Maynard, James},
  title = {Small gaps between primes},
  journal = {Annals of Mathematics},
  volume = {181},
  pages = {383--413},
  year = {2015}
}

@article{polymath8b,
  author = {Polymath, D. H. J.},
  title = {Variants of the {S}elberg sieve, and bounded intervals containing many primes},
  journal = {Research in the Mathematical Sciences},
  volume = {1},
  pages = {12},
  year = {2014}
}

Citing this problem

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OP-00918, "The Hardy–Littlewood prime k-tuples conjecture", clopen.solutions.