Dense clusters of primes in short intervals and Chebotarev sets: uniformity in the cluster size m and distribution estimates with exponential log-scale savings

Maynard's 'Dense clusters of primes in subsets' (Compositio Mathematica, 2016) proves that any set of primes that is well distributed in arithmetic progressions contains many primes that are close together: under an explicit distribution hypothesis (Hypothesis 1), an admissible set of $k$ linear forms takes at least $c\delta\log k$ prime values, where $\delta$ measures the density of the prime set among the values of the forms, at more than a constant times $\#\mathcal{A}(x)/((\log x)^k e^{Ck})$ points of $\mathcal{A}$. The applications to short intervals (Theorem 3.4: clusters of $m$ primes among $k=\exp(O(m))$ forms inside $[x,x+y]$, $y\ge x^{7/12+\epsilon}$) and to Chebotarev sets (Theorem 3.5) are, however, stated for fixed $m$ only, with ineffective constants, because the underlying Bombieri-Vinogradov-type distribution results (Timofeev's short-interval estimate and the Murty-Murty Chebotarev variant) save only a fixed power of $\log x$. Maynard remarked that estimates saving $\exp(-c\sqrt{\log x})$ after excluding bad moduli 'would allow uniformity for $m\le\epsilon\log\log x$', and speculated that adapting the hypothesis to weighted sums would extend Theorem 3.4 to $y\ge x^{0.525}$. The $x^{0.525}$ extension was proved by Alweiss and Luo (2018), and the anticipated bad-moduli exclusion mechanism was realized in the global long-interval setting by Ford-Green-Konyagin-Maynard-Tao; but the uniformity-in-$m$ problem in the two settings actually posed (short intervals and Chebotarev sets), the effectivity of the constants, and the conjecturally correct bounds of order $y/(\log x)^m$ all remain open.

Background

A central theme of analytic number theory is that conclusions about primes follow from information about their distribution in arithmetic progressions. The Bombieri-Vinogradov theorem states that, for every $A>0$, the total error made by the prime number theorem for arithmetic progressions, when averaged (with a maximum over residue classes $a$ coprime to $q$) over all moduli $q$ at most $x^{1/2-\epsilon}$, is bounded by a constant times $x/(\log x)^A$; the strength of such a statement is measured both by the range of moduli and by how small the error term is relative to $x$. The conjectural truth, of the size accessible under the generalized Riemann hypothesis for Dirichlet $L$-functions, would allow error terms such as $x\exp(-c\sqrt{\log x})$, which is smaller than $x/(\log x)^A$ for every fixed $A$. Unconditionally, $\exp(-c\sqrt{\log x})$-strength is known only for very small moduli (the Siegel-Walfisz theorem), and its obstruction for larger moduli is the possible existence of an exceptional real character (a Landau-Siegel zero); the Landau-Page theorem shows there is at most one such bad modulus below $\exp(c\sqrt{\log x})$, so statements saving $\exp(-c\sqrt{\log x})$ are expected to hold after excluding one modulus.

A set $L=\{L_1,\dots,L_k\}$ of integer linear functions $L_i(n)=a_i n+b_i$ is called admissible if the product $L_1(n)\cdots L_k(n)$ has no fixed prime divisor (that is, for every prime $p$ there is an integer $n_p$ with $p\nmid L_1(n_p)\cdots L_k(n_p)$); admissibility is necessary for several of the $L_i(n)$ to be prime simultaneously, and Dickson's prime $k$-tuples conjecture predicts that for an admissible set about $x/(\log x)^k$ values of $n\le x$ make all $k$ forms prime. A construction going back to Goldston, Pintz and Yildirim, transformed by Maynard, shows unconditionally (using only Bombieri-Vinogradov-level distribution) that for admissible $L$ with $k$ large, there are many $n$ for which at least $c\log k$ of the forms $L_i(n)$ are prime; this is the engine behind bounded gaps between primes.

Maynard's 2016 paper 'Dense clusters of primes in subsets' made this engine applicable to restricted sets: given a set of integers $\mathcal{A}$ and a set of primes $\mathcal{P}$, Hypothesis 1 asks that (1) $\mathcal{A}$ is well distributed in arithmetic progressions with total error a small negative power of $\log x$ over moduli up to $x^{\theta}$, (2) the primes of $\mathcal{P}$ lying among the values $L(\mathcal{A})$ are likewise well distributed in most progressions, and (3) $\mathcal{A}$ is not concentrated in a single progression; under Hypothesis 1, Theorem 3.1 produces, at more than a constant times $\#\mathcal{A}(x)/((\log x)^k\exp(Ck))$ points $n$ of $\mathcal{A}$, at least $c\delta\log k$ prime values among the $L_i(n)$, where $\delta$ measures the density of $\mathcal{P}$ in $L(\mathcal{A})$.

The applications include: strings of $m$ consecutive primes all congruent to $a\bmod q$ and of diameter at most $\epsilon\log x$, uniformly for $m$ up to a constant times $\log\log x$ and $q$ at most $(\log x)^{1-\epsilon}$ (Theorem 3.3, building on Shiu and Freiberg); dense clusters in short intervals: for fixed $m$ and $\epsilon>0$, with $k=\exp(O(m))$ forms, at least $y/(\log x)^k$ values of $n$ in $[x,x+y]$ give at least $m$ prime values, whenever $y\ge x^{7/12+\epsilon}$ (Theorem 3.4, using Timofeev's 1987 Bombieri-Vinogradov-type theorem in short intervals, which holds in the same range); and the same for primes whose Artin symbol (the Frobenius conjugacy class) in a fixed Galois extension $K/\mathbb{Q}$ lies in a prescribed conjugacy class $C$, where the distribution input is the 1987 Chebotarev variant of Bombieri-Vinogradov due to Murty and Murty, valid for moduli up to $x^{\theta}$ with $\theta<\min(1/2,2/|G|)$ (Theorem 3.5). Two features of Theorems 3.4 and 3.5 motivated the present problem. First, $m$ is fixed: the threshold $x_0$ and the implied constant depend on $m$, because the available distribution results save only an arbitrary power of $\log x$, while the sieve error terms for $k$ forms deteriorate like $(\log x)^{-100k^2}$; once $k=\exp(Cm)$ is allowed to reach a power of $\log x$ (which is what $m$ comparable to $\log\log x$ requires), one needs error savings of scale $\exp(-c\sqrt{\log x})$. Second, the implied constants are ineffective, again because of possible exceptional characters. Maynard states that one would presume the distribution results 'can be extended to save $\exp(-c\sqrt{\log x})$ or similar (having excluded some possible bad moduli), which would allow uniformity for $m\le\epsilon\log\log x$', and that adapting Hypothesis 1 to weighted sums 'could use presumably' the results of Harman-Watt-Wong and Kumchev to extend Theorem 3.4 to the wider range $x^{0.525}\le y\le x$; he also notes that the lower bound $y/(\log x)^k$ is smaller than the expected truth $y/(\log x)^m$ by a fixed power of $\log x$, a loss he calls 'an unavoidable feature of the method when looking at bounded length intervals'.

After 2016, Alweiss and Luo carried out the $x^{0.525}$ extension; Ford-Green-Konyagin-Maynard-Tao realized the bad-moduli exclusion mechanism with effective $\exp(-c\sqrt{\log x})$-savings, but only for long intervals of length at least $x$; Polymath and Stadlmann improved the analogous $\exp(Cm)$-type sieve losses for the global small-gaps problem; and Guth and Maynard improved the prime number theorem in short intervals (uniformly, at exponents beyond $17/30$) without producing an arithmetic-progression distribution analogue of that strength. None of this addresses the uniformity in $m$ for the localized settings of Theorems 3.4 and 3.5.

Problem Statement

The problem is to remove the fixed-$m$ and ineffective-constant restrictions of Maynard's localized dense-cluster theorems, whose common scientific objective is to bring the short-interval setting (Theorem 3.4) and the Chebotarev setting (Theorem 3.5) to the parameter uniformity in the cluster size $m$ that the general Theorem 3.1 permits and that the global results of Ford-Green-Konyagin-Maynard-Tao have already realized for long intervals. A solution of either of the following two targets solves the problem; establishing both closes it in full.

(A) Short intervals. Prove that there exist absolute constants $C>0$ and $\epsilon_0>0$ such that the following holds. For every $0<\epsilon\le\epsilon_0$, every $x\ge x_1(\epsilon)$ (the threshold $x_1$ may depend on $\epsilon$, but not on $m$, $k$, $y$, or the forms), all real $y$ with $x^{0.525}\le y\le x$, every positive integer $m\le\epsilon\log\log x$ with $k=\exp(Cm)$, and every admissible set $L=\{L_1,\dots,L_k\}$ of linear functions $L_i(n)=a_i n+b_i$ with $1\le a_i\ll(\log x)^{1/\epsilon}$ and $0\le b_i\ll x$, one has

$$\#\{n\in[x,x+y]:\ \text{at least }m\text{ of }L_1(n),\dots,L_k(n)\text{ are prime}\}\ \gg\ \frac{y}{(\log x)^k},$$

with an implied constant that is absolute — independent of $m$, $k$, $x$, $y$, and $L$ (dependence on $\epsilon$ is permitted, as for the threshold $x_1(\epsilon)$) — and effectively computable.

(B) Chebotarev sets. Prove the analogous uniform version of Theorem 3.5: for every Galois extension $K/\mathbb{Q}$ with discriminant $\Delta_K$ and every conjugacy class $C$ of its Galois group, there are constants $C_K,\epsilon_K>0$ depending only on $K$ such that for every $0<\epsilon\le\epsilon_K$, every $x\ge x_1(K,\epsilon)$ (the threshold may depend only on $K$ and $\epsilon$, not on $m$ or the forms), every positive integer $m\le\epsilon\log\log x$ with $k=\exp(C_Km)$, and every admissible set $L=\{L_1,\dots,L_k\}$ of linear functions $L_i(n)=a_i n+b_i$ with $1\le a_i\ll(\log x)^{1/\epsilon}$, $0\le b_i\ll x$, and $(a_i,\Delta_K)=1$ for all $i$, one has

$$\#\{x\le n\le 2x:\ \text{at least }m\text{ of }L_1(n),\dots,L_k(n)\text{ are prime with Artin symbol }C\}\ \gg\ \frac{x}{(\log x)^{\exp(C_Km)}},$$

with an implied constant depending only on $K$ and $\epsilon$ (not on $m$ or the forms) and effectively computable. (The source states Theorem 3.5 for each fixed admissible set, with the threshold $x_0$ depending on the set and with no coefficient-size restriction beyond $(a_i,\Delta_K)=1$; a uniform version must let the set vary with $m$, so the coefficient ranges of Theorem 3.4 are adopted here.)

Deeper strengthenings noted in the source, acknowledged to face the parity barrier (Selberg's fundamental obstruction that purely sieve-theoretic methods cannot distinguish primes from numbers with an odd number of prime factors) and therefore not required for the completion boundary above: strengthen the lower bounds toward the conjecturally correct order $y/(\log x)^m$ in place of $y/(\log x)^k$ with $k=\exp(O(m))$, i.e. remove the fixed power-of-$\log x$ loss for bounded length intervals, and reduce the exponential dependence of $k$ on $m$ (cf. the improvements of Polymath and Stadlmann in the global setting).

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