OP-00919 Open

The Hardy–Littlewood asymptotic formula for Goldbach representations

Let $N_2(n)$ be the number of representations of $n$ as a sum of two primes, order counted, and let $$ C_2 = \prod_{q \ge 3} \left(1 - \frac{1}{(q-1)^2}\right), $$ the product running over the odd primes. Hardy and Littlewood's method, which gives the corresponding result for three primes on a hypothesis generalising the Riemann hypothesis, leads to the formula below, but they could not complete the proof even assuming that no Dirichlet $L$-function has a zero with real part greater than $\tfrac12$.

Problem. Prove that every large even number is the sum of two odd primes, and that, as $n \to \infty$ through even numbers, $$ N_2(n) \sim 2 C_2 \frac{n}{(\log n)^2} \prod_{p \mid n,\ p > 2} \frac{p-1}{p-2}. $$

Source: G. H. Hardy and J. E. Littlewood, Partitio Numerorum III, Acta Math. 44 (1923), Conjecture A; a formula of this kind was first suggested by Sylvester (1871).

Imported from doi.org/10.1007/BF02403921.

Classification

Primary: NT — Number Theory
Keywords: additive prime number theory, Goldbach's conjecture, Hardy–Littlewood conjectures, 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{sylvester1871,
  author = {Sylvester, J. J.},
  title = {On the partition of an even number into two primes},
  journal = {Proceedings of the London Mathematical Society},
  volume = {4},
  pages = {4--6},
  year = {1871}
}

@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{montgomery1975,
  author = {Montgomery, H. L. and Vaughan, R. C.},
  title = {The exceptional set in {G}oldbach's problem},
  journal = {Acta Arithmetica},
  volume = {27},
  pages = {353--370},
  year = {1975}
}

@article{oliveiraesilva2014,
  author = {Oliveira e Silva, Tom{\'a}s and Herzog, Siegfried and Pardi, Silvio},
  title = {Empirical verification of the even {G}oldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$},
  journal = {Mathematics of Computation},
  volume = {83},
  pages = {2033--2060},
  year = {2014}
}

Citing this problem

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OP-00919, "The Hardy–Littlewood asymptotic formula for Goldbach representations", clopen.solutions.