The Hardy–Littlewood asymptotic conjecture for the binary Goldbach representation function

For an even integer $2n$ , let $G(2n)$ denote the number of ways to write $2n$ as a sum of two primes, counted with order (so $2n=p+p'$ and $2n=p'+p$ are distinct representations, and $2n=p+p$ is allowed). In 1923 Hardy and Littlewood conjectured the asymptotic formula $G(2n)\sim 2\,C_2\,N_2(n)\,\frac{2n}{(\log 2n)^2}$ , where $C_2\approx 0.66016$ is the twin-prime constant and $N_2(n)=\prod_{p\mid n,\,p>2}\frac{p-1}{p-2}$ is a singular-series factor depending on the odd prime divisors of $n$ . A century later this remains one of the central open problems of analytic number theory: no unconditional asymptotic, and not even an unconditional lower bound of the conjectured order with the singular series, is known for individual even integers — even $G(2n)\ge 1$ for all large $n$ , the binary Goldbach conjecture itself, is open. The problem is to prove the conjectured main term unconditionally, for all even integers without restriction.

Background

Goldbach's 1742 conjecture asserts that every even integer greater than 2 is the sum of two primes; for an even $N=2n$ write $G(2n)$ for the number of such representations (counting representations with attention paid to order, as in Hardy–Littlewood 1923: $2n=p+p'$ and $2n=p'+p$ are distinct, and $2n=p+p$ is allowed). A simple density heuristic — two independent primes of size about $n$ each occur with probability roughly $1/(\log n)^2$ — suggests $G(2n)$ should be of order $n/(\log n)^2$ , but the prime factors of $n$ bias the count: the Hardy–Littlewood conjecture of 1923, developed through the circle method (a technique that converts counting problems about additive representations into integrals of generating functions over the unit circle), predicts the precise asymptotic

G(2n)\;\sim\;2\,C_2\,N_2(n)\,\frac{2n}{(\log 2n)^2},\qquad C_2=\prod_{p>2}\Bigl(1-\frac{1}{(p-1)^2}\Bigr)\approx 0.66016,\quad N_2(n)=\prod_{p\mid n,\,p>2}\frac{p-1}{p-2},

where $C_2$ is the twin-prime constant and the singular-series factor $N_2(n)$ encodes the enhancement when $n$ has few odd prime factors (for example $2n$ divisible by 3 can use pairs $p,\,2n-p$ both avoiding the residue class of $n$ mod 3, giving the factor 2 when $3\mid n$ ). Hardy and Littlewood also proved, assuming the Generalized Riemann Hypothesis (GRH — the conjecture that all non-trivial zeros of Dirichlet $L$ -functions, which encode the distribution of primes in residue classes, lie on the line of real part 1/2), that all sufficiently large odd numbers are sums of three primes; that ternary statement was later made unconditional, and Helfgott (2013) proved the ternary Goldbach conjecture for every odd number greater than 5 outright. The binary (two-prime) problem is of a structurally different, harder nature. The unconditional partial results are of three kinds, none approaching the pointwise main term. First, density results: work in the 1930s showed that almost all even integers are sums of two primes, and Montgomery and Vaughan (1975) proved that the exceptional set $E(x)$ of even integers up to $x$ failing the representation satisfies $E(x)\ll x^{1-\delta}$ for some $\delta>0$ — but this says nothing about any individual $n$ . Second, Chen's theorem (1973): every sufficiently large even integer is $p+P_2$ , where $P_2$ denotes an integer with at most two prime factors — the celebrated $1+2$ result, still the closest qualitative approximation to binary Goldbach. Third, upper-bound and average results: sieve and exponential-sum methods give upper bounds for $G(2n)$ of the conjectured order — recently sharpened by Lichtman (2023), whose level of distribution $66/107$ for primes (the largest known proportion $\theta$ for which primes are uniformly distributed in residue classes to moduli up to $x^{\theta}$ , exceeding the square-root barrier $\theta=1/2$ ) yields the best known upper bounds for Goldbach representations — while under the Riemann Hypothesis (RH, the GRH for the single Riemann zeta function) the average behavior is precisely understood: Languasco and Zaccagnini (2010) proved under RH an explicit formula for the summatory von Mangoldt-weighted Goldbach function $\sum_{n\le N}R(n)=\frac{N^2}{2}-2\sum_\rho\frac{N^{\rho+1}}{\rho(\rho+1)}+O(N\log^3 N)$ , and Goldston and Yang (2016) obtained RH-conditional asymptotics for smoother averages. Recent work continues to circle the pointwise statement without reaching it: Alsetri and Shao (2024) proved density versions (subsets of the primes of relative density above 1/2 still represent almost all even integers), and Chirre and Valås Hagen (2025) proved under RH that every interval $(x,\,x+123\log^2 x]$ with $x\ge 2$ contains a Goldbach number (an even integer that is itself a sum of two primes). The obstruction is the parity barrier: sieve methods, which dominate all unconditional approaches, cannot distinguish primes from products of two primes at the available level of distribution $1/2$ , while the pointwise asymptotic would require exactly such parity-sensitive information about prime sums.

Problem Statement

Prove, without any unproved hypothesis, the Hardy–Littlewood asymptotic for the binary Goldbach representation function in its original unrestricted generality: as $n\to\infty$ through the positive integers,

G(2n)\;\sim\;2\,C_2\,N_2(n)\,\frac{2n}{(\log 2n)^2},\qquad C_2=\prod_{p>2}\Bigl(1-\frac{1}{(p-1)^2}\Bigr),\quad N_2(n)=\prod_{p\mid n,\,p>2}\frac{p-1}{p-2},

where $G(2n)$ is the number of representations of $2n$ as a sum of two primes, counted with order as above. Equivalently, with the von Mangoldt function $\Lambda$ (which weights a prime power $p^k$ by $\log p$ ), $\sum_{k=2}^{2n-2}\Lambda(k)\Lambda(2n-k)\sim 2C_2N_2(n)\cdot 2n$ , since the unweighted and weighted main terms determine each other by partial summation. The conjecture is asserted for all even integers without restriction — not merely for almost all $n$ , not along a subsequence, not under a density condition, and not conditional on zero hypotheses. A refutation of the asymptotic is likewise an admissible resolution: an explicit unbounded family of even integers whose certified representation counts deviate from the main term by a fixed multiplicative factor.

References

G. H. Hardy and J. E. Littlewood, 'Some problems of "Partitio numerorum"; III: On the expression of a number as a sum of primes', Acta Mathematica 44 (1923), 1–70. DOI: 10.1007/BF02403921, https://doi.org/10.1007/BF02403921
Chen, Jing-run, 'On the representation of a larger even integer as the sum of a prime and the product of at most two primes', Scientia Sinica 16 (1973), 157–176. zbMATH: 3499888, https://zbmath.org/3499888
H. L. Montgomery and R. C. Vaughan, 'The exceptional set of Goldbach's problem', Acta Arithmetica 27 (1975), 353–370. DOI: 10.4064/aa-27-1-353-370, https://doi.org/10.4064/aa-27-1-353-370
H. A. Helfgott, 'The ternary Goldbach conjecture is true', arXiv:1312.7748 (2013). DOI: 10.48550/arXiv.1312.7748, https://arxiv.org/abs/1312.7748
Alessandro Languasco and Alessandro Zaccagnini, 'The number of Goldbach representations of an integer', arXiv:1011.3198 (2010). DOI: 10.48550/arXiv.1011.3198, https://arxiv.org/abs/1011.3198
D. A. Goldston and Liyang Yang, 'The Average Number of Goldbach Representations', arXiv:1601.06902 (2016). DOI: 10.48550/arXiv.1601.06902, https://arxiv.org/abs/1601.06902
Jared Duker Lichtman, 'Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier', arXiv:2309.08522 (2023). DOI: 10.48550/arXiv.2309.08522, https://arxiv.org/abs/2309.08522
Ali Alsetri and Xuancheng Shao, 'Density versions of the binary Goldbach problem', arXiv:2405.18576 (2024). DOI: 10.48550/arXiv.2405.18576, https://arxiv.org/abs/2405.18576
Andrés Chirre and Markus Valås Hagen, 'On Goldbach numbers in short intervals', arXiv:2512.23534 (2025). DOI: 10.48550/arXiv.2512.23534, https://arxiv.org/abs/2512.23534