The maximal well-factorable level of distribution of primes in arithmetic progressions

In a fixed residue class, Maynard (2020) proved that the primes of size $x$ are equidistributed in arithmetic progressions, on average over moduli, up to level $x^{3/5-\epsilon}$ when weighted by triply well-factorable weights, and up to $x^{7/12-\epsilon}$ for the well-factorable upper-bound weights of the linear sieve, while remarking that $3/5$ appeared to be the limit of the method and leaving the sieve applications as future work. The subsequent literature resolved the source's central question of whether the exponent can be pushed beyond $3/5$: Lichtman (2023) raised the triply well-factorable level to $x^{66/107}$, and to $x^{5/8}$ assuming Selberg's eigenvalue conjecture; Pascadi (2025) then made $x^{5/8-\epsilon}$ unconditional, improved the linear-sieve weight level to $x^{3/5-\epsilon}$, and deduced the current twin-prime bound $\pi_2(x)\le(3.203+o(1))\Pi_2(x)$. The surviving open problem is the core of the source question: raise the well-factorable level of distribution of the primes beyond $x^{5/8-o(1)}$ for triply well-factorable weights, beyond $x^{4/7}$ for doubly well-factorable weights (unchanged since 1986), and beyond $x^{3/5}$ for the linear sieve weights; determine the optimal exponents or the obstructions to them; and, in the limit, decide whether the Elliott-Halberstam range $x^{1-o(1)}$ is attainable in this weighted sense. Current methods appear capped at $5/8$, the missing ingredient being large sieve inequalities of mixed additive-multiplicative type for exceptional Maass forms.

Background

Write $\pi(x;q,a)$ for the number of primes $p\le x$ with $p\equiv a\pmod q$, and $\pi(x)$ for the number of primes up to $x$. The prime number theorem for arithmetic progressions describes $\pi(x;q,a)$ for each fixed modulus $q$, and the Generalized Riemann Hypothesis (GRH) would describe it uniformly for moduli up to about $x^{1/2}$. Unconditionally, the Bombieri-Vinogradov theorem reaches the same range on average over moduli: for every $A>0$,

$\sum_{q\le x^{1/2}/(\log x)^B}\ \sup_{(a,q)=1}\Big|\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\Big|\ \ll_A\ \frac{x}{(\log x)^A},$

with $B=B(A)$, where $\varphi$ is Euler's totient function. The largest admissible power $\vartheta$ in $q\le x^{\vartheta}$ is called the level (or exponent) of distribution of the primes, and it is the key quantitative input to sieve arguments about primes. The Elliott-Halberstam conjecture asserts that $\vartheta=1-\epsilon$ is admissible for every $\epsilon>0$; it remains one of the central open problems of analytic number theory, and for unrestricted moduli no value $\vartheta>1/2$ is known for this unweighted, worst-residue statement (variants restricted to smooth or conveniently factorable moduli do exceed $1/2$; see Maynard's Part I and Stadlmann's refinement for smooth moduli).

Many sieve applications need less: a fixed residue class $a$ (such as $a=1$ or $a=2$), with the average weighted by sequences $(\lambda_q)$ that arise as sieve weights. Bombieri-Friedlander-Iwaniec (1986) introduced well-factorable weights: $(\lambda_q)$, of level $Q$, is well-factorable if for every factorization $Q=Q_1Q_2$ there exist sequences bounded by $1$, supported on $q_i\le Q_i$, with $\lambda_q=\sum_{q_1q_2=q}\gamma^{(1)}_{q_1}\gamma^{(2)}_{q_2}$. Their celebrated Theorem 10 gives equidistribution of the primes with such weights up to level $x^{4/7-\epsilon}$, which stood as the record for primes in any setting for over three decades. Maynard's 2020 paper (the source of this problem) defined triply well-factorable weights — the analogous three-sequence decomposition required for every factorization $Q=Q_1Q_2Q_3$ — and reached level $x^{3/5-\epsilon}$ with them, remarking that $3/5$ appeared to be the limit of the current method and that imposing still stronger factorability on the weights seemed to give no further gain. The application-driven case is the linear sieve, the dimension-one combinatorial sieve whose upper-bound weights of level $D$ produce twin-prime-type estimates; a well-factorable variant of these weights due to Iwaniec behaves essentially identically in sieve problems, and Maynard reached level $x^{7/12-\epsilon}$ for it, proved via a factorization proposition for the sieve support that he showed, by explicit configurations of prime factors, to be sharp at $7/12$.

All these results follow Linnik's dispersion method: decompose the primes using Heath-Brown's identity (a finite combinatorial identity for the prime-counting weight), apply Cauchy-Schwarz to convert the average over moduli into bilinear sums, and bound the resulting sums of Kloosterman sums — complete exponential sums $S(m,n;c)=\sum_{d\bmod c,\,(d,c)=1}e((md+n\overline{d})/c)$, where $e(t)=e^{2\pi i t}$ and $\overline{d}$ denotes the multiplicative inverse of $d$ modulo $c$ — by the spectral theory of automorphic forms, via the Kuznetsov trace formula (an identity relating sums of Kloosterman sums to the spectral data of automorphic forms), in the form of large sieve inequalities for Maass forms (non-holomorphic eigenfunctions of the hyperbolic Laplacian on congruence subgroups) due to Deshouillers-Iwaniec. The quality of the attainable exponent is governed by the exceptional spectrum — Maass forms with Laplace eigenvalue below $1/4$, writing the eigenvalue as $\lambda=1/4-\theta^2$, so that $\theta\in[0,7/32]$ by the Kim-Sarnak bound and $\theta=0$ would be Selberg's eigenvalue conjecture, which is open. A parallel story holds for smooth numbers (integers with no prime factor above $y$), where Drappeau (2015) reached level $x^{3/5-\epsilon}$ and Pascadi (2025) later reached $x^{5/8-\epsilon}$.

The state after the source paper is as follows. Lichtman (2023) raised the triply well-factorable level to $x^{66/107}\approx x^{0.617}$ unconditionally and to $x^{5/8}$ assuming Selberg's eigenvalue conjecture, and Pascadi (2025) made level $x^{5/8-\epsilon}$ unconditional — combining new large sieve inequalities for exceptional Maass forms with a mean-value theorem of Watt — while also improving the linear-sieve weight level to $x^{3/5-\epsilon}$. These levels feed directly into sieve records: the current twin-prime upper bound $\pi_2(x)\le(3.203+o(1))\Pi_2(x)$, where $\Pi_2(x)=\frac{2x}{(\log x)^2}\prod_{p>2}\frac{1-2/p}{(1-1/p)^2}$ is the Hardy-Littlewood prediction, and Goldbach-type estimates that for the first time use a level of distribution beyond the square-root barrier $x^{1/2}$, giving the greatest improvement on those bounds since 1966. Pascadi states that $5/8$ is now the best exponent available 'in essentially any setting relevant for sieve theory', that no more flexible weighting is known to help with current methods, and that the missing input is a large sieve inequality for sequences of mixed additive and multiplicative structure. The doubly well-factorable level has remained $x^{4/7-\epsilon}$ since 1986.

Sharpness remarks attached to the factorization propositions in Maynard's and Pascadi's papers show their factorization systems to be optimal at the corresponding levels, so the question of the optimal exponent recurs at each new level.

Problem Statement

The problem is to raise, and to determine the optimal value of, the level of distribution of the primes in arithmetic progressions with well-factorable-type weights — the largest $\vartheta\in(0,1)$ such that for every $A,\epsilon>0$ and every fixed nonzero integer $a$,

$\sum_{\substack{q\le x^{\vartheta-\epsilon}\\(q,a)=1}}\lambda_q\Big(\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\Big)\ \ll_{a,A,\epsilon}\ \frac{x}{(\log x)^A},$

where $(\lambda_q)$ is a sequence of level $Q=x^{\vartheta-\epsilon}$ which is (i) well-factorable in the sense of Bombieri-Friedlander-Iwaniec (for every factorization $Q=Q_1Q_2$ a decomposition $\lambda_q=\sum_{q_1q_2=q}\gamma^{(1)}_{q_1}\gamma^{(2)}_{q_2}$ with 1-bounded sequences supported on $q_i\le Q_i$), or (ii) triply well-factorable in the sense of Maynard (the analogous three-sequence decomposition for every $Q=Q_1Q_2Q_3$), or (iii) the well-factorable upper-bound weights of the linear sieve of level $Q$ (Iwaniec's variant, as treated in Maynard's Theorem 1.2 and Pascadi's Theorem 1.3(ii)), or (iv) another explicitly defined class of sieve weights in the same hierarchy (e.g. quadruply well-factorable weights, or Lichtman's programmably factorable modifications of the linear sieve weights). The moduli are unrestricted — not confined to smooth, prime, or otherwise special moduli — and the residue class is fixed, with the implied constant allowed to depend on it.

The current records to beat are $\vartheta=4/7$ for class (i) (Bombieri-Friedlander-Iwaniec, 1986, never improved), $\vartheta=5/8$ for class (ii) (Pascadi, 2025, unconditional), and $\vartheta=3/5$ for class (iii) (Pascadi, 2025; Lichtman's modified weights of class (iv) reach $10/17$). The targets are:

(T1) prove the displayed estimate for some $\vartheta>5/8$ in class (ii), or $\vartheta>3/5$ in class (iii), or $\vartheta>4/7$ in class (i), with the weight class not silently restricted (it must retain the standard sieve applicability of its class, and any restriction to special moduli changes the problem and does not answer this one);

(T2) determine the supremal admissible $\vartheta$ in each class, or characterize the obstructions;

(T3) in the limit, decide whether the Elliott-Halberstam range $\vartheta=1-o(1)$ is attainable in this weighted sense.

These targets are nested milestones of the single objective above, not interchangeable alternatives. Any one of the three alternatives in (T1) is a self-contained partial advance and settles a definite sub-claim on its own; the problem as a whole is resolved by (T2), which determines the supremal admissible exponent in each class, or the obstruction that caps it, with (T3) its limiting case. Class (iv) is an eligible but optional setting: a solution may be posed there, but the problem does not require it.

The single common objective underlying these targets is the maximal on-average equidistribution of primes in arithmetic progressions with sieve-friendly weights; the level parameter is the same object across classes (i)-(iv), and the standard downstream sieve consequences (twin-prime-type upper bounds, Chen-type and Goldbach-type estimates, shifted-prime factorizations) are linked corollaries of the same level rather than separate objectives. An answer that only derives new sieve consequences from the existing $x^{5/8}$ level does not solve this problem.

References

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