OP-00920 Open

The Hardy–Littlewood asymptotic formula for prime pairs with a given difference

For a positive even integer $k$, let $P_k(n)$ be the number of pairs of primes $p,\ p' = p + k$ less than $n$, and let $C_2 = \prod_{q \ge 3} \bigl(1 - (q-1)^{-2}\bigr)$, the product running over the odd primes. Hardy and Littlewood note that the formula below predicts about equal numbers of prime pairs differing by $2$ and by $4$, and about twice as many differing by $6$, in good agreement with counts up to $5000$.

Problem. Prove that for every even $k$ there are infinitely many prime pairs $p,\ p + k$, and that $$ P_k(n) \sim 2 C_2 \frac{n}{(\log n)^2} \prod_{p \mid k,\ p > 2} \frac{p-1}{p-2} \qquad (n \to \infty). $$

Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture B.

Imported from doi.org/10.1007/BF02403921.

Classification

Primary: NT — Number Theory
Keywords: Hardy–Littlewood conjectures, prime gaps, singular series, twin primes
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{chen1973,
  author = {Chen, Jing Run},
  title = {On the representation of a larger even integer as the sum of a prime and the product of at most two primes},
  journal = {Scientia Sinica},
  volume = {16},
  pages = {157--176},
  year = {1973}
}

@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-00920, "The Hardy–Littlewood asymptotic formula for prime pairs with a given difference", clopen.solutions.