Kuperberg 2023 sums singular series large sets tail

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conjecture_1_3: States the Hardy–Littlewood k-tuples conjecture with one power-saving error uniform over k up to (log log x) cubed and over admissible tuples in [0, (log x) squared]; the hypothesis assumed by the 2026 conditional claims on problem 251, with no unconditional support.

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V. Kuperberg, Sums of singular series with large sets and the tail of the distribution of primes, Q. J. Math. 74 (2023), no. 4, 1457--1479, doi:10.1093/qmath/haad030; arXiv:2210.09775 (v1 18 October 2022; v2 15 June 2023, 20 pages). The journal version is paywalled and was not consulted; the journal data are as cited by the manuscripts that assume the conjecture (see below). The arXiv record lists the license CC BY 4.0.

The retained folder-name PDF is arXiv v2, the version that the 2026 manuscripts on problem 251 cite by number. Provenance: fetched from https://arxiv.org/pdf/2210.09775v2 on 2026-09-17 (UTC), 259,004 bytes. The PDF is LaTeX-generated with a text layer; page numbers below are the PDF's own (physical page = printed page). The arXiv record (https://arxiv.org/abs/2210.09775, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Contents

Throughout, $\mathcal H=\{h_1,\dots,h_k\}$ is a set of $k$ distinct integers, $\nu_{\mathcal H}(p)$ the number of residue classes modulo $p$ it occupies, and

$$\mathfrak S(\mathcal H)=\prod_{p}\frac{1-\nu_{\mathcal H}(p)/p}{(1-1/p)^k}$$

its singular series (equation (2), p. 1).

• Theorem 1.1 (p. 2): for fixed $\delta>1/2$ and $k=O((\log h)^{1-\delta})$, the sum $T_k(h)$ of $\mathfrak S(h_1,\dots,h_k)$ over distinct $h_1,\dots,h_k\le h$ equals $h^k+O(h^{k-\beta})$ for some $\beta=\beta(\delta)>0$; this extends Gallagher's average (3) to growing $k$.
• Theorem 1.2 (p. 2): with no condition on the growth of $k$, $T_k(h)\ll h^k\prod_{p\le k^3}(1-1/p)^{-k}\ll h^k(3\log k)^k$.
• Conjecture 1.3 (p. 3): the Hardy–Littlewood $k$-tuples conjecture in a form uniform in $k\le(\log\log x)^3$ and in admissible tuples inside $[0,(\log x)^2]$, with a power-saving error.
• Theorem 1.4 (p. 3) and Corollaries 1.5--1.6 (p. 4): under Conjecture 1.3, for $h=\lambda\log x$ and $r\ll(\log h)^{1-\delta}$ with some $\delta>1/2$, the $r$th moment of $\pi(n+h)-\pi(n)$ over $n\le x$ is the Poisson moment $\sum_{\ell}\left\{{r\atop\ell}\right\}\lambda^{\ell}$ up to a factor $1+o(1)$; by Corollary 1.5, if $\lambda$ is nondecreasing, $k\ll(\log h)^{1-\delta}$ and $k/(\lambda+1)\to\infty$, the number of $n\le x$ with at least $k$ primes in $(n,n+h]$ is $\ll x\exp(-k/(\lambda e))$ for $\lambda\ge1$ (and $\ll x\exp(-k/((\lambda+1)e))$ otherwise); Corollary 1.6 gives a weaker bound with no growth condition on $k$.
• Conjectures 1.7 (p. 4) and 1.10 (p. 5): for $\lambda=o((\log x)^\varepsilon)$ for every $\varepsilon>0$ and $k\ll(\log h)^2$, the number of $n\le x$ with exactly $k$ primes in $(n,n+h]$ is asymptotic to the Poisson prediction $x\lambda^ke^{-\lambda}/k!$ (1.7) and is $\ll x\exp(-k/(\lambda e))$ (1.10).
• Theorem 1.8, Corollary 1.9 (pp. 4--5) and Theorem 4.1 (p. 15): unconditional moment and tail bounds from the Selberg sieve, with Theorem 4.1 giving $#{n\le x:\ n+h_i\text{ prime for all }i}\le(2+\varepsilon)^k k!,\mathfrak S(\mathcal H),x/\log^k x$ up to a relative error, for $k=o((\log x)^{1/4})$.

Only the statements above were read, from the text layer and (for Conjecture 1.3) on the page image of p. 3; no proof was checked and nothing here is independently reviewed.

Relation to the catalog

The paper proves nothing about a catalog problem. Its Conjecture 1.3 is the hypothesis assumed by the two 2026 conditional claims on Problem 251: Land's Theorem 2 assumes it in a large-$x$ form, and Ringer's Corollary 1.2 derives its averaged one-sided hypothesis from the conjecture's equation (7). Tao's 2023 paper on Problem 15, filed as tao_2023_convergence_alternating_series_erdos_assuming_hardy, restates the conjecture with the wider range $k\le(\log\log x)^5$; that card, not this one, carries the Problem 15 account. No unconditional result of this uniformity is known: the paper itself reports only small computer tests (p. 3), and even the fixed-$k$ Hardy–Littlewood conjecture is open for $k\ge2$.

Bears on. #251, as the assumed hypothesis of claimed conditional results only; the paper contains no result on the problem.