Completely uniform distribution of primes to moduli $x^{1/2+\delta}$: the balanced Type II estimate, the four-factor terms, and the full extension of Bombieri–Vinogradov

The Bombieri–Vinogradov theorem equidistributes the primes up to $x$ among residue classes, on average over moduli up to $x^{1/2}$, with complete uniformity in the residue class. Maynard's memoir "Primes in arithmetic progressions to large moduli III: Uniform residue classes" gave the first results beyond the $x^{1/2}$ barrier with comparable residue-class uniformity, for moduli of size $x^{1/2+\delta}$ having conveniently sized divisors — but with decisive compromises: the completely uniform version (Theorem 1.1) has only a weak error term, the good-error version (Theorem 1.2) constrains the residue classes, and the remaining version (Theorem 1.3) works with a minorant of the primes. The paper identifies the obstructions: no Type II (bilinear) estimate uniform over residue classes covers the balanced range $N,M\approx x^{1/2}$, and the terms of the Heath-Brown identity with four factors each of length $x^{1/4+O(\delta)}$ cannot be handled without restricting the residue classes. The problem is to establish the full, completely uniform extension of Bombieri–Vinogradov to moduli $x^{1/2+\delta}$ with suitable factorization, and the problem is open: no result is known that achieves residue-class-uniform distribution of primes to moduli beyond $x^{1/2}$ with a strong error term.

Background

The distribution of primes among residue classes is measured by $\pi(x;q,a)$, the number of primes $p\le x$ with $p\equiv a\pmod q$ and $(a,q)=1$; the prime number theorem for arithmetic progressions gives $\pi(x;q,a)\sim\pi(x)/\varphi(q)$ for fixed $q$, where $\varphi$ is Euler's totient function. Many sieve-theoretic applications need this asymptotic only on average over the moduli, and the foundational result of that kind is the Bombieri–Vinogradov theorem: for every $A>0$ there is a constant $B=B(A)$ such that

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

where the supremum over all reduced residue classes makes the estimate completely uniform in the class $a$. This is an unconditional substitute, on average, for the Generalized Riemann Hypothesis, and $x^{1/2}$ (the square-root barrier) is the limit of what is known for unrestricted moduli. The Elliott–Halberstam conjecture predicts the same bound with moduli up to $x^{1-\epsilon}$ for every fixed $\epsilon>0$ and remains one of the central open problems of analytic number theory.

Passing the barrier at all, even in restricted settings, has a long history. In the 1980s Bombieri, Friedlander and Iwaniec proved distribution results for moduli of size $x^{1/2+\delta}$ with well-factorable weights or conveniently sized factors, but their methods — which ultimately rely on bounds for sums of Kloosterman sums (the exponential sums $S(a,b;c)=\sum_{x\bmod c}^{*}e((ax+b\bar x)/c)$ over invertible residue classes $x\bmod c$, where $e(t)=e^{2\pi i t}$ and $\bar x$ is the multiplicative inverse of $x$ modulo $c$, estimated via the spectral theory of automorphic forms) — forced the residue class $a$ to be a single fixed small integer. Work culminating in Zhang's proof of bounded gaps between primes produced the first beyond-$x^{1/2}$ estimates with better residue-class uniformity, namely for smooth moduli (all prime factors small) with residue classes constrained by a polynomial congruence, relying on Weil- and Deligne-type bounds from algebraic geometry (sharp estimates for complete exponential sums over finite fields) instead of spectral methods; the Polymath project then refined this to an estimate in which one residue class $a$ is shared by all moduli. Such estimates are the key input for the sieve method of Goldston–Pintz–Yıldırım that yields bounded gaps between primes, and the residue-class uniformity is exactly what that method needs.

Maynard's trilogy "Primes in arithmetic progressions to large moduli" I–III separates the two difficulties. Part I treats a fixed residue class: the expected asymptotic for $\pi(x;q,a)$ holds for all but $O(\delta Q)$ moduli $q\sim Q=x^{1/2+\delta}$ with a convenient factor, and, when the modulus has a divisor of size about $x^{1/21}$, for moduli as large as $x^{11/21-\epsilon}$. Part II treats well-factorable weights (roughly, weights $\lambda_q$ that factor as Dirichlet convolutions of bounded sequences on dyadic ranges, the weights arising in the linear sieve; later authors use a threefold 'triply-well-factorable' analogue), reaching moduli $x^{3/5-\epsilon}$. Part III is the first work to achieve, for moduli beyond $x^{1/2}$, uniformity over residue classes comparable to that of Bombieri–Vinogradov: its estimates consider $\exp(x^{1/2+\delta+o(1)})$ collections of residue classes — more than any previous beyond-$x^{1/2}$ result, and comparable to what the classical theorem would consider if extended to moduli of that size.

Part III proves three main theorems, each with a decisive compromise. Theorem 1.1 gives complete residue-class uniformity for moduli $q=q_1q_2$ with a factor of size close to $x^{1/10}$, but only with a weak error term (it saves only a factor $O(\delta)$ over the trivial bound). Theorem 1.2 gives a full power-of-logarithm error term for moduli $q=q_1q_2q_3=x^{1/2+\delta}$ with the factors in explicit ranges, but the residue class $a\pmod{q_1q_2q_3}$ is constrained to lie in a class $b\pmod{q_1q_2}$ that does not depend on $q_3$ — a relaxation of complete uniformity obtained by adapting Zhang's method. Theorem 1.3 proves complete uniformity with a good error term, but only for a minorant $\rho$ of the prime indicator (with $\sum_{n\le x}\rho(n)\ge\pi(x)/8$), not for the primes themselves.

The engine behind these theorems is the standard reduction of prime-counting to bilinear and multilinear estimates. A combinatorial identity of Heath-Brown (which decomposes the prime indicator into finitely many convolution terms) reduces the task to estimating convolutions $\alpha_n\beta_m$, and their multilinear analogues, in residue classes with essentially arbitrary coefficients: "Type I" estimates concern one long factor, and "Type II" (bilinear) estimates concern two factors of comparable size, with one coefficient sequence $\alpha$ satisfying a Siegel–Walfisz condition (the source's Definition 1: for all $d,q\ge1$ with $(a,q)=1$, the sum of $\alpha_n$ over $n\equiv a\pmod q$ with $(n,d)=1$ equals $1/\varphi(q)$ times the sum over the terms coprime to $dq$, up to an error $\ll_A N\tau(d)^{O(1)}/(\log N)^A$, where $\tau$ is the divisor function; the condition is satisfied by the indicator functions of the primes and of the squarefree numbers). Maynard's Proposition 5.1 is a Type II estimate of this kind for moduli $qr$ with $QR=x^{1/2+\delta}$, uniform over residue classes, but valid only in unbalanced ranges excluding both factors having size $x^{1/2}$; Proposition 5.2 extends the range somewhat but still fails to cover the balanced case. Two specific configurations are identified as the obstructions: genuinely balanced convolutions with $N\asymp M\asymp x^{1/2}$, and the terms of the Heath-Brown identity with four factors each of length $x^{1/4+O(\delta)}$. In the source's own words, the failure to produce a completely uniform estimate for the balanced terms "is why Theorem 1.1 and Theorem 1.2 fail to give a full extension of the Bombieri-Vinogradov Theorem to moduli $q\sim x^{1/2+\delta}$ with suitable factorization properties"; for the four-factor terms the author imposes "some slight constraints on the residue classes", which is exactly what limits Theorem 1.2. The problem is to remove these obstructions.

Problem Statement

Fix a sufficiently small $\delta>0$ and let $x\to\infty$. The problem is to establish the missing completely uniform extension of the Bombieri–Vinogradov theorem to moduli beyond the square-root barrier, in the factorable range considered by Maynard's "Primes in arithmetic progressions to large moduli III", together with the bilinear and multilinear estimates that obstruct it.

The objective is a mean-value theorem of the following strength: for moduli $q$ of size $x^{1/2+\delta}$ possessing suitably sized factors, in the sense of the factorization conditions of Maynard's Theorems 1.1–1.3, the estimate being required for each of the three factorization classes of those theorems: the class of Theorem 1.1 ($q=q_1q_2$ with a divisor $q_1\le x^{1/10-3\delta}/(\log x)^C$ for a sufficiently large absolute constant $C$, and $q_2\le x^{4/10+4\delta}(\log x)^C$), the class of Theorem 1.2 ($q=q_1q_2q_3=x^{1/2+\delta}$ with $x^{40\delta}<q_2<x^{1/20-7\delta}$ and $x^{1/10+12\delta}/q_2<q_3<x^{1/10-4\delta}/q_2^{3/5}$), and the class of Theorem 1.3 ($q=q_1q_2$ with a divisor $q_1\in[x^{2/5+5\delta},x^{3/7}]), one has

$$\sum_{q\ \mathrm{allowed}}\ \sup_{(a,q)=1}\Bigl|\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)}\Bigr|\ \ll_{A,\delta}\ \frac{x}{(\log x)^A}$$

for every fixed $A>0$, where the implied constant may depend on $A$ and on the fixed $\delta$, and where the supremum is over all reduced residue classes: the estimate must be completely uniform in the residue classes — treating $\exp(x^{1/2+\delta+o(1)})$ collections of residue classes, as many as the Bombieri–Vinogradov theorem would treat if extended to moduli of that size — and the error term must be of Bombieri–Vinogradov quality (a power of the logarithm, as in Theorem 1.2 of the source, not the weak error term of Theorem 1.1, which is $O(\delta\pi(x))$ and so saves only a factor $O(\delta)$ over the trivial bound). The same statement with the von Mangoldt-weighted counting function in place of $\pi(x;q,a)$ — that is, counting prime powers $n\le x$ with $n\equiv a\pmod q$ weighted by the von Mangoldt function $\Lambda(n)$, which equals $\log p$ on powers of a prime $p$ and is zero elsewhere — is equivalent. The statement must be unconditional; ineffective constants are tolerated, as in the source, but no unproved spectral, algebraic, or zero hypothesis may be assumed.

Two estimates are identified in the source as the obstructions:

(i) a bilinear (Type II) estimate, uniform over residue classes, covering the balanced range: for $QR=x^{1/2+\delta}$ with $q\sim Q$, $r\sim R$ satisfying factorization conditions in the sense of the source's Propositions 5.1–5.2 (for example a factor of size about $x^{1/10}$), for complex sequences with $|\alpha_n|,|\beta_m|\le\tau(n)^A$ (where $\tau$ is the divisor function) and $\alpha$ satisfying a Siegel–Walfisz condition, and with $NM\asymp x$, prove

$$\sum_{q\sim Q}\ \sum_{r\sim R}\ \sup_{(a,qr)=1}\Bigl|\sum_{m\sim M}\ \sum_{n\sim N}\alpha_n\beta_m\Bigl(\mathbf 1_{nm\equiv a\ (\mathrm{mod}\ qr)}-\frac{\mathbf 1_{(nm,qr)=1}}{\varphi(qr)}\Bigr)\Bigr|\ \ll_A\ \frac{x}{(\log x)^A},$$

in ranges that include the balanced case $N\asymp M\asymp x^{1/2}$. The source's Proposition 5.1 stops at $N\le x^{1/2-3\delta}$ up to logarithmic factors, and Proposition 5.2 covers only $N$ near $x^{2/5}$; and

(ii) the analogous uniform multilinear estimate covering the terms of the Heath-Brown identity with four factors each of length $x^{1/4+O(\delta)}$, which the source can handle only after imposing constraints on the residue classes.

The common objective is the mean-value theorem: the completely uniform Bombieri–Vinogradov-type statement for primes to moduli $x^{1/2+\delta}$ with suitable factorization and a power-of-logarithm error term, for each of the three factorization classes above. The broader Elliott–Halberstam conjecture (moduli up to $x^{1-\epsilon}$ without factorization restrictions) is explicitly outside this problem, as it was outside the scope of the source; so are the source's secondary technical tasks of making the constants effective and optimizing the numerical admissible ranges.

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