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Source. Conjecture 1.3, p. 3 of arXiv:2210.09775v2, read on the page image and the text layer. The conjecture is stated, not proved; the paper proves nothing about it beyond the small computer tests reported on the same page.

Statement

The paper writes $1_{\mathcal P}$ for the indicator function of the primes, $\Lambda$ for the von Mangoldt function, $\nu_{\mathcal H}(p)$ for the number of residue classes modulo $p$ occupied by $\mathcal H=\{h_1,\dots,h_k\}$,

$$\mathfrak S(\mathcal H)=\prod_{p}\frac{1-\nu_{\mathcal H}(p)/p}{(1-1/p)^k}, \qquad \operatorname{li}_k(x)=\int_2^x\frac{dy}{(\log y)^k},$$

and calls $\mathcal H$ admissible when $\nu_{\mathcal H}(p)<p$ for every prime $p$.

Conjecture 1.3 (Hardy–Littlewood $k$-tuples conjecture, uniform version). "There exist two absolute constants $\epsilon>0$ and $C>0$ such that for all $x$, for all $k\le(\log\log x)^3$, and for all admissible tuples $\mathcal H=\{h_1,\dots,h_k\}\subset[0,(\log x)^2]$,

$$\Bigl|\sum_{n\le x}1_{\mathcal P}(n+h_1)\cdots1_{\mathcal P}(n+h_k) -\mathfrak S(\mathcal H)\operatorname{li}_k(x)\Bigr|\le Cx^{1-\varepsilon}.$$

Equivalently, for possibly different values of $\varepsilon$ and $C$,

$$\Bigl|\sum_{n\le x}\Lambda(n+h_1)\cdots\Lambda(n+h_k) -\mathfrak S(\mathcal H)x\Bigr|\le Cx^{1-\varepsilon}."$$

The first display is the paper's equation (7) and the second its equation (8); the paper writes $\epsilon$ in the preamble and $\varepsilon$ in the displays.

The quantifier order matters: one pair $(\varepsilon,C)$ serves every $x$, every $k$ in the range and every admissible tuple in the window. The ordinary Hardy–Littlewood conjecture, one asymptotic for each fixed tuple, does not supply this uniformity. The paper adds (p. 3) that computer tests for several sets of size $k=10$ and $x\le5500$ found errors below $(\log x)^6x^{1/2}$, and that the range of $k$ and $h$ "is also likely possible to extend", while "it is difficult to say when Hardy–Littlewood convergencee [sic] should break down."

Versions used by other sources

• Tao (arXiv:2308.07205, Conjecture 1.3, p. 2; the card tao_2023_convergence_alternating_series_erdos_assuming_hardy) cites this conjecture and states it for all $x\ge10$, all $k\le(\log\log x)^5$ and all tuples of distinct integers in $[0,\log^2x]$, remarking that restricting to admissible tuples is unnecessary because the bound is easy when $\mathfrak S(\mathcal H)=0$, and that the exponent $5$ replaces Kuperberg's $3$ "for technical reasons." The phrase "$x\ge10$" is Tao's; the original says "for all $x$".
• Land's 2026 manuscript (Conjecture 1, form (K), p. 1; the card land_2026_conditional_proof_irrationality_prime_series) uses a "large-$x$ form": constants $\varepsilon,C>0$ such that for every sufficiently large $x$ and every admissible $A\subseteq[0,(\log x)^2]\cap\mathbb Z$ with $1\le|A|\le(\log\log x)^3$, $|\sum_{1\le m\le x}\prod_{a\in A}1_{\mathcal P}(m+a) -\mathfrak S(A)\operatorname{li}_{|A|}(x)|\le Cx^{1-\varepsilon}$. Its Lean predicate UniformHardyLittlewoodConjecture quantifies $\varepsilon$, $C$ and a threshold $x_0$ before $x$ and $A$.
• Ringer's 2026 manuscript (equation (22), p. 19; the card ringer_2026_local_gap_statistics_telescoping_normality) quotes (7) for admissible distinct $E\subset[0,(\log x)^2]$ with $|E|\le(\log\log x)^3$ and derives from it the averaged one-sided hypothesis $(\mathrm{AHL}_\kappa)$ of its Theorem 1.1 (Section 5.4), noting that a uniform error $O_A(x\exp\{-A(\log\log x)^2\})$ for tuples of order at most $(A/2)\log\log x$ would already suffice for its qualitative conclusions.

Standing

A conjecture. No unconditional theorem of this uniformity is known; the fixed-$k$ Hardy–Littlewood conjecture itself is open for every $k\ge2$. Tao remarks (p. 2 of arXiv:2308.07205) that the conjecture "has been verified almost surely" for the random sifted model of the primes of Banks, Ford and Tao, which is evidence about the model, not about the primes. Every result that assumes this conjecture is conditional, and its acceptance would not make the assumed statement true.

Bears on. #251, as the hypothesis of the claimed conditional results of Land (Theorem 2) and Ringer (Corollary 1.2); it is not a result on the problem.