The sharp upper-bound constant for twin primes: reducing Selberg's factor 8 (1947) toward the Hardy–Littlewood conjecture

Selberg's four-page 1947 note 'On an elementary method in the theory of primes' introduced the upper-bound sieve with square divisor-sum weights and announced, as an application, that the number of twin primes up to $x$ is at most $10.6\,x/\log^2 x$ for all sufficiently large $x$ — eight times the density predicted by the Hardy–Littlewood conjecture — with the full derivation, the handling of remainder terms, an announced lower-bound method, and extensions to other sequences all deferred to later publications. The deferred details were subsequently published and the Selberg sieve became a foundational tool of number theory, while the constant 8 was reduced in an unbroken line of work: 6 (Pan 1964), 4 (Bombieri–Davenport 1966), 3.9171 (Chen 1978), 3.5 (Bombieri–Friedlander–Iwaniec 1986), 3.29956 (Lichtman 2022), 3.2290 (Lichtman 2023), and most recently 3.203 (Pascadi 2025), all in units of the Hardy–Littlewood prediction $\Pi_2(x)$. The open problem is the quantitative core surviving from that note: determine the optimal constant $K$ in $\pi_2(x)\le(K+o(1))\Pi_2(x)$ — any unconditional $K<3.203$, and ultimately the conjectured $K=1$ — in the face of the parity problem of sieve theory, the recognized structural obstruction that keeps sieve-type methods from detecting the primality of $p+2$.

Background

Two integers $p$ and $p+2$ that are both prime are called twin primes; write $\pi_2(x)=\#\{p\le x:\ p,\ p+2\ \text{prime}\}$. Hardy and Littlewood (1923) conjectured the quantitative asymptotic $\pi_2(x)\sim\Pi_2(x)$ as $x\to\infty$, where

$$\Pi_2(x)=\frac{2x}{(\log x)^2}\prod_{p>2}\frac{1-2/p}{(1-1/p)^2}\approx\frac{1.3203\,x}{(\log x)^2}$$

(the product over primes $p>2$ is the twin-prime constant $\approx 0.66016$; some authors fold the leading factor $2$ into it). Whether $\pi_2(x)\to\infty$ at all is the twin prime conjecture, open for over a century; the upper-bound side asks how close to $\Pi_2(x)$ one can prove that $\pi_2(x)$ must lie. Brun (1919), with his combinatorial sieve (an iterated inclusion–exclusion procedure for counting integers with no small prime factor), was the first to show that twin primes are scarce: $\pi_2(x)$ is bounded by an absolute constant multiple of $x/(\log x)^2$ — the correct order of magnitude, but with a large unspecified constant — so in particular the sum of reciprocals of the twin primes converges. The source paper is Selberg's four-page 1947 note, which introduced a new elementary device: if $A$ is a finite sequence of integers and one has a multiplicative density model $\#\{a\in A:\ \rho\mid a\}=N/f(\rho)+R_\rho$ (with $N=|A|$, $f$ multiplicative, and a remainder $R_\rho$), then for arbitrary real weights $\lambda_\nu$ with $\lambda_1=1$ supported on the divisors of $P(z)$ (the product of the primes below $z$),

$$\#\{a\in A:\ (a,P(z))=1\}\ \le\ \sum_{a\in A}\Big(\sum_{\nu\mid a}\lambda_\nu\Big)^2;$$

expanding the square and minimizing the resulting quadratic form in the $\lambda_\nu$ yields an upper bound (the Selberg, or $\Lambda^2$, upper-bound sieve). Applied to the sequence $A=\{p+2:\ p\le x\ \text{prime}\}$ this gave, as stated in the note without full derivation, $\pi_2(x)<10.6\,x/(\log x)^2$ for $x\ge x_0$ — i.e. $\pi_2(x)\le(8+o(1))\Pi_2(x)$: the constant at last explicit, though eight times the conjectured density, with the threshold $x_0$ and the treatment of the remainders $R_\rho$ deferred to a later publication, and a corresponding lower-bound method only mentioned. Selberg developed the general method in his 1949 Trondheim lecture (which also stressed the method's limitations) and his 1950 ICM address; the $\Lambda^2$ sieve became one of the two standard engines of sieve theory (the other being the combinatorial linear sieve, the modern refinement of Brun's inclusion–exclusion approach; see Friedlander–Iwaniec 2010), systematized in the monographs of Halberstam–Richert, Greaves, Iwaniec–Kowalski, and Friedlander–Iwaniec. The modern theory of the twin-prime upper bound combines three ingredients. First, the level of distribution of the primes — the exponent $\theta$ up to which the primes are equidistributed among residue classes on average: by the Bombieri–Vinogradov theorem (1965), this holds up to moduli $x^{1/2-\varepsilon}$, and with a level $x^{\theta-\varepsilon}$ available for the weighted remainder sums the sieve produces, the standard machinery gives $\pi_2(x)\le(2/\theta+o(1))\Pi_2(x)$; going beyond $\theta=1/2$ (the square-root barrier) requires deep estimates for bilinear forms — Linnik's dispersion method and the Deshouillers–Iwaniec spectral theory of Kloosterman sums (complete exponential sums modulo an integer $q$ that control the error terms in the distribution of primes) — as in Fouvry–Iwaniec, Bombieri–Friedlander–Iwaniec, Lichtman and Pascadi. Second, the combinatorial linear sieve with well-factorable remainder weights (weights whose moduli may be split into a product of two smaller moduli, each within the range accessible to the available distribution estimates). Third, the switching principle introduced in Chen's celebrated 1973 theorem that infinitely many primes $p$ have $p+2$ equal to a product of at most two primes — the parity-limited substitute for a lower bound on twin primes; the principle replaces the sifted sequence by an auxiliary one in the hard cases and transfers the resulting bound back. The recognized structural obstruction is the parity problem of sieve theory: sieves cannot distinguish integers with an odd number of prime factors from those with an even number, so no pure sieve can prove the twin prime conjecture, and the conjectured upper-bound constant $1$ is likewise out of reach of pure sieve arguments; every improvement of the constant below 4 imports non-sieve equidistribution inputs. Against this background, the question surviving from Selberg's note is: what is the smallest constant $K$ for which $\pi_2(x)\le(K+o(1))\Pi_2(x)$ can be proved — the current record being $K=3.203$ (Pascadi 2025), still more than three times the conjectured value?

Problem Statement

Let $\pi_2(x)=\#\{p\le x:\ p\ \text{and}\ p+2\ \text{both prime}\}$ and let $\Pi_2(x)=\frac{2x}{(\log x)^2}\prod_{p>2}\frac{1-2/p}{(1-1/p)^2}$ denote the Hardy–Littlewood predicted count of twin primes ($\approx 1.3203\,x/(\log x)^2$). Selberg's elementary method gives $\pi_2(x)\le(8+o(1))\Pi_2(x)$ (the 1947 form: $\pi_2(x)<10.6\,x/(\log x)^2$ for $x\ge x_0$), and the best unconditional bound at the time of writing is $\pi_2(x)\le(3.203+o(1))\Pi_2(x)$ (Pascadi 2025). The problem is to reduce the constant below the current record: prove unconditionally that $\pi_2(x)\le(K+o(1))\Pi_2(x)$ for an explicit $K<3.203$, with the ultimate target the conjectured sharp bound $K=1$ — that is, for every $\varepsilon>0$, $\pi_2(x)\le(1+\varepsilon)\Pi_2(x)$ for all sufficiently large $x$. Proving a lower bound $\pi_2(x)\to\infty$ (the twin prime conjecture itself) is not part of this problem.

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