Level of distribution of primes in arithmetic progressions with well-factorable weights beyond 5/8, and the twin-prime and Goldbach sieve constants

Lichtman (2023) proved that the primes are equidistributed in arithmetic progressions with triply well-factorable weights to moduli up to $x^{66/107}$ (about $x^{0.617}$) unconditionally and $x^{5/8}$ under Selberg's eigenvalue conjecture, and used a residue-uniform level of $153/256$ to give the first Goldbach upper bound beyond the square-root distribution barrier: $G(a)\lesssim3.3907\Pi_a(a)$, alongside $\pi_2(x)\lesssim3.2290\Pi_2(x)$ for twin primes, where $\Pi_a,\Pi_2$ are the Hardy–Littlewood predicted main terms. Pascadi (2025) then removed the eigenvalue hypothesis, making the triply-well-factorable level $5/8-o(1)$ unconditional and improving the twin-prime constant to $3.203$. The open problem is how far these marks can be pushed toward the Elliott–Halberstam range $\vartheta<1$: prove a level of distribution strictly beyond $5/8$ with triply well-factorable weights (fixed or small residue), or beyond $153/256$ uniformly in the residue $|a|<x^{1+\epsilon}$ (the Goldbach setting); any such level advance correspondingly improves the sieve constants $3.203$ and $3.3907$. No level advance beyond these marks is known, so the problem is open.

Background

Distribution of primes in arithmetic progressions. Write $\pi(x;q,a)$ for the number of primes $p\le x$ with $p\equiv a\pmod q$, and $\varphi(q)$ for the number of integers up to $q$ coprime to $q$. The prime number theorem for arithmetic progressions gives $\pi(x;q,a)\approx\pi(x)/\varphi(q)$, and the quality of this approximation as the modulus $q$ grows is a central measure of our understanding of the primes. The Bombieri–Vinogradov theorem (1965) shows, on average over moduli $q\le x^{\vartheta}$, GRH-strength error terms for every $\vartheta<1/2$; $\vartheta$ is called the level of distribution of the primes, and $\vartheta=1/2$ is the square-root barrier. The Elliott–Halberstam conjecture (1970), among the most consequential open problems in analytic number theory, predicts the same averaged statement for every level $\vartheta<1$; it is the conjectural endpoint of this problem, and even the unweighted theorem at some level $1/2+\delta$ remains open.

Well-factorable weights. Many sieve arguments need the averaged equidistribution not with trivial weights but with weights $\lambda_q$ that themselves arise in the sieve. A sequence $\lambda_q$ is well-factorable of level $Q$ (in the sense used by Bombieri–Friedlander–Iwaniec, 1986; Lichtman, arXiv:2309.08522, Definition 1.3) if, for any factorization $Q=Q_1Q_2$ with $Q_1,Q_2\ge1$, there exist $1$-bounded sequences $\gamma^{(1)},\gamma^{(2)}$ supported on $q_i\le Q_i$ such that $\lambda=\gamma^{(1)}*\gamma^{(2)}$ (divisor convolution), i.e. $\lambda_q=\sum_{q=q_1q_2}\gamma^{(1)}_{q_1}\gamma^{(2)}_{q_2}$; it is triply well-factorable (Lichtman, arXiv:2309.08522, Definition 1.4; Pascadi, arXiv:2505.00653, Definition 1.1, quoting Maynard) if, for any factorization $Q=Q_1Q_2Q_3$ with $Q_1,Q_2,Q_3\ge1$, there exist $1$-bounded sequences $\gamma^{(1)},\gamma^{(2)},\gamma^{(3)}$ supported on $q_i\le Q_i$ such that $\lambda_q=\sum_{q=q_1q_2q_3}\gamma^{(1)}_{q_1}\gamma^{(2)}_{q_2}\gamma^{(3)}_{q_3}$. Heuristically, every modulus in the support can be split into two (respectively three) factors of nearly arbitrary sizes. This flexibility is what lets the analysis tolerate moduli beyond $x^{1/2}$. Bombieri–Friedlander–Iwaniec (1986) proved a level $4/7$ for well-factorable weights and a fixed nonzero residue $a$; Maynard (2020, published 2025) raised this to $3/5$ and treated uniform residue classes in a companion memoir. The weights produced by the (upper-bound) linear sieve are of this type, so such estimates feed directly into upper bounds for twin primes and Goldbach representations.

Methods and the spectral barrier. The proofs proceed by the dispersion method: a finite combinatorial identity for the von Mangoldt function (the prime-counting weight) reduces the problem to bilinear sums in progressions; Poisson summation and Cauchy–Schwarz rearrangements reduce these to sums of Kloosterman sums (exponential sums of the shape $e(a\bar{m}n/q)$ over residues), which are bounded through the Kuznetsov trace formula relating Kloosterman sums to Fourier coefficients of automorphic cusp forms, via the spectral large sieve of Deshouillers–Iwaniec (1982). A key obstruction is the exceptional spectrum: Maass cusp forms with near-real spectral parameter. Selberg's eigenvalue conjecture asserts there are none ($\theta=0$); the best known bound is $\theta\le7/32$ (Kim–Sarnak, 2003). The dependence of the final level on $\theta$, and on the residue $a$, was sharpened by Drappeau–Pratt–Radziwiłł (2023) and Assing–Blomer–Li (2021), and then optimized by Lichtman.

The source results. Lichtman (2023, arXiv:2309.08522, appendix by Drappeau) proved level $\vartheta=(5-4\theta)/(8-6\theta)$ for triply well-factorable weights: $66/107\approx0.617$ unconditionally, $5/8=0.625$ under Selberg's eigenvalue conjecture; and uniformly over residues $0<|a|<x^{1+\epsilon}$ the level $\vartheta_1=(5-\theta)/8=153/256\approx0.5977$. Via his modified linear sieve (published 2025), these give $\pi_2(x)\lesssim3.2290\Pi_2(x)$, where $\pi_2(x)$ counts primes $p\le x$ with $p+2$ also prime, and $G(a)\lesssim3.3907\Pi_a(a)$ for the number $G(a)$ of representations of an even integer $a$ as a sum of two primes, where $\Pi_a(x)=2S_a x/(\log x)^2$ and $S_a$ is the Hardy–Littlewood singular series (an explicit Euler product, with $\Pi_2$ the case of twin primes); the Hardy–Littlewood conjecture of 1923 predicts the true ratios tend to $1$, while pure sieve methodology cannot cross the ratio $2$ (the parity barrier), so $2$ is the natural sieve-limit target. The Goldbach constant improved the 2004 record $3.9104$ (Wu) by $13\%$, the first major move since the constant $4$ of Bombieri–Davenport (1966); earlier constants descend from $8$ (1947) and $3.9171$ (1978). In the sources the twin-prime and Goldbach constant bounds are downstream applications of the level-of-distribution results, and these constants are the classical quantitative measure of progress in this area.

State of the art after 2023. Pascadi (2024) proved new large sieve inequalities for exceptional Maass forms; Pascadi (2025) combined these with the spectral machinery to show that both primes and smooth numbers (integers with no large prime factor) are equidistributed in progressions to moduli $x^{5/8-o(1)}$ with triply well-factorable weights, unconditionally and for fixed residue — eliminating the Selberg-eigenvalue hypothesis at $5/8$ — and improved the twin-prime constant to $3.203$. Two structural facts frame the open problem: assuming Selberg's conjecture no longer directly improves the $5/8$ record; and for large residues $a\lesssim x^{1+\epsilon}$ (the Goldbach-relevant range) the spectral obstruction is replaced by Ramanujan–Petersson-type dependence at the primes dividing $a$, still governed by the Kim–Sarnak bound $7/32$. Lichtman also observed that quadruply well-factorable weights appear to give no further advantage, suggesting the modulus-factorization vein of the dispersion method is optimized near $5/8$ without a genuinely new idea.

Problem Statement

The problem is to extend the primes' level of distribution in arithmetic progressions beyond the square-root barrier $x^{1/2}$, past the current records, toward the Elliott–Halberstam range $\vartheta<1$, on three linked fronts sharing one objective — measuring how far the primes can be proved equidistributed among residue classes in the settings that sieve methods consume.

(F) Fixed or small residue, triply well-factorable weights. Prove that for some fixed $\delta>0$, for every $A>0$ and every nonzero residue $a$ (fixed, or uniformly for $|a|<x^{\epsilon}$), all triply well-factorable weights $\lambda_q$ of level $x^{5/8+\delta}$ satisfy

$$\Big|\sum_{\substack{q\le x^{5/8+\delta}\\ (q,a)=1}}\lambda_q\Big(\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\Big)\Big|\ll \frac{x}{(\log x)^A}.$$

Current record: level $5/8-o(1)$, unconditional (Pascadi 2025). More generally: determine the supremal admissible level in $(5/8,1)$.

(U) Uniform residue — the Goldbach-relevant setting. Prove that for some fixed $\vartheta_1>153/256\approx0.5977$ and every $A>0$, all triply well-factorable weights $\lambda_q$ of level $x^{\vartheta_1}$ satisfy

$$\sup_{0<|a|<x^{1+\epsilon}}\Big|\sum_{\substack{q\le x^{\vartheta_1}\\ (q,a)=1}}\lambda_q\Big(\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\Big)\Big|\ll \frac{x}{(\log x)^A},$$

uniformly over all residues $0<|a|<x^{1+\epsilon}$. Current record: $\vartheta_1=153/256$ (Lichtman 2023, whose Theorem 1.7 displays exactly this estimate at $\vartheta_1=(5-\theta)/8-\epsilon$).

(C) Quantitative sieve benchmarks. Improve the best known constants in

$$\pi_2(x)\le (C_2+o(1))\Pi_2(x),\qquad G(a)\le (C_G+o(1))\Pi_a(a)$$

beyond the current records $C_2=3.203$ (twin primes up to $x$) and $C_G=3.3907$ (Goldbach representations of sufficiently large even $a$), toward the sieve-limit value $2$; the Hardy–Littlewood conjectures predict the true ratios tend to $1$.

The problem in its full extent is closed only by reaching every level $\vartheta<1$ with well-factorable weights — the Elliott–Halberstam endpoint — under which the constants would approach $2$.

References

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