On the Hardy--Littlewood--Chowla conjecture

Guangshi L\"u; Jie Wu; Zhiwei Wang

REVIEW

On the Hardy--Littlewood--Chowla conjecture

Guangshi Lü, Zhiwei Wang, Jie Wu

Number Theory · Submitted 14 Oct 2026 · 14 references

Pith's verdict: First unconditional claim of cancellation of μ on a fixed shifted prime sequence, with an elegant weight-averaging finish; the Type II core rests on an unpublished OpenAI preprint and an AI-assisted root-counting lemma that the manuscript does not let a referee certify.

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CitationGuangshi L\"u, Jie Wu, Zhiwei Wang. "On the Hardy--Littlewood--Chowla conjecture." arXiv:2610.20242 (2026). https://pith.science/paper/Z7Y2KNSF

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Claim paper

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Editor's note

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The thing you should know: this paper claims the first unconditional cancellation of the Möbius function on a fixed shifted prime sequence — Σ_{p≤x} μ(p+2) = o(π(x)) — and I could not find an explicit error in it. Neither could the automated reader. But the load-bearing Type II section is built from an operator framework adapted from an unpublished OpenAI preprint ([11]), and its decisive input, Lemma A.1, sits in an appendix the acknowledgements credit to ChatGPT. Conditional is the right verdict.

What is actually new checks out. Hildebrand states the conjecture, Lichtman has averages over shifts, Lichtman–Teräväinen handle all-but-o(H) shifts; nobody has a fixed shift. The auxiliary weight device (Proposition 2.3) is the nicest piece in the paper and it is checkable: build weights with mean 1 and variance O(1/b), prove the weighted estimate, then send b to infinity. The Type I section is careful Bombieri–Vinogradov work, Proposition 6.1 is a textbook Siegel–Walfisz argument, and the final elimination of the weight is clean.

Where it is soft. The verification gap is concentrated in Lemma A.1's uniform root count at precision min{d0YM,N}^{−1/100} and in the operator estimates (4.51) of Section 4.4. The appendix states and proves A.1, but the proof adapts unpublished [11, Lemma 3.3], and that uniformity is exactly what Proposition 4.4's cubing argument consumes: if the exponent or the S-power in the error weakens, the L^{−2AR} saving fails and the Type II bound collapses. I walked through the Kloosterman fourth-moment step, the summation over u, and the involution for N < d0YM and found no contradiction — this is a certification gap, not a located error. The structural weakness is that a result of this strength rests its central mechanism on a source few readers can inspect.

Two things we should not count against the paper: there is no circularity (the weights are eliminated by b→∞, no fitted constants), and the restriction to shift 2 is acknowledged (Remark 1). I would add that the latter is not purely cosmetic: the parity conjugacy in Proposition 4.3 exploits the even shift, so odd shifts do not follow from this argument as written.

Bottom line: this deserves a serious referee — several, I think, charged specifically with Lemma A.1 and the debt to [11]. I would ask the authors to make the borrowed operator construction self-contained or to hold the paper until [11] is public. Not a desk reject, not an accept on this version.

Referee report

Significance

If correct, this is a breakthrough: it would resolve a folklore conjecture going back at least to Hildebrand (1989), give the first unconditional cancellation of the Möbius function on a fixed shift of the primes, and provide a model-case resolution of a parity-problem obstruction. The Type I estimate and the squarefree-density argument (Proposition 6.1) are transparent and standard, and the paper is commendably explicit about the ordering of parameter choices (A0, K, E0) and about the fact that constants may depend on K and ρ. The main caveat is that the central Type II estimate is not independently checkable from the manuscript as submitted: Proposition 4.2 and (4.51) rest on an operator construction explicitly modeled on unpublished reference [11], and Lemma A.1, the input to Proposition 4.4, is stated and proved in the appendix in a way that the authors themselves say follows [11]. I found no explicit contradiction in the internal steps I checked, but the decisive estimate is a certification gap rather than a demonstrated error.

Major objections (2)

[§4.4, §4.5, and Appendix A] The Type II estimate is gated by Lemma A.1 and by the auxiliary operator bounds leading to (4.51). These are not independently verifiable from the manuscript: §4.4 introduces the operators G_j and E_j with the phrase 'Similar to the construction of [11, Sections 4.1–4.3]', and Appendix A states that the proof of Lemma A.1 'follows the argument of [11]'. Reference [11] is an unpublished OpenAI preprint rather than a refereed or archived mathematical source, and the acknowledgements further state that ChatGPT helped establish Appendix A. Since Proposition 4.4 needs the uniformity of Lemma A.1 at precision L^{-2AR} for arbitrary fixed A, and since (4.52), Proposition 4.2, and hence Theorem 1 collapse if that uniformity fails, the paper should either provide a fully self-contained proof of the operator estimates and of Lemma A.1 that does not route through [11], or cite and make available a published/archived version of [11] whose proofs have been independently checked. As submitted, the central claim is conditional on an unverifiable source.
[§4.3 and (4.32)] In Proposition 4.4, the passage from the finite-residue count (A.5) to the normalized error estimate (4.32) is compressed. Lemma A.1 has error O(d0YMN S^2 min{d0YM,N}^{-1/100}), and the application involves moduli S that can be as large as exp(O_K(L^{0.96})) and path coefficient masses exp(O_K(L^{0.73})); the displayed bound E_count ≪ d0YMN x^{-5cδ/8+o(1)} is asserted after absorbing all discrete choices. The saving is plausible, but the quantitative bookkeeping should be written out, because the final L^{-2AR} saving depends on the power saving x^{-cδ} dominating exp(O_K(L^{0.96}+L^{0.73})) and on the error being uniform over the Bonferroni terms. This is a load-bearing transition and needs to be checkable line by line.

Minor objections (4)

[Abstract and Remark 1] The abstract says 'we prove this conjecture' for the shifted-prime Möbius sum, but Theorem 1 is proved only for the fixed shift h_1=2; Remark 1 acknowledges this, but the abstract and title-level statements should be qualified to avoid overclaiming the generality of the result.
[§2.2] The sentence 'Note that by dyadic decomposition, and it suffices to consider the sums appearing in Theorems 1 and 2 over the interval (x,2x]' is grammatically incomplete; it should be rewritten, and the dyadic reduction for Theorem 2 should be stated explicitly.
[§4.2, condition (4.15)] The symbol Z is used both for the sifting parameter in (2.1) and for a product of primes in condition (4.15). The text notes this collision, but changing one of the notations would substantially improve readability of the good-state conditions.
[References] Reference [11] should be flagged as an unpublished preprint in the references and cited as such in the text; as it now appears, a reader could mistake it for a standard published source.

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The paper

The paper claims

This paper proves the Möbius function cancels over primes shifted by two — $\sum_{p\le x}\mu(p+2)=o(\pi(x))$ — settling a 1989 folklore conjecture for this shift and forcing each sign to occur with density $1/2$.

The reading

An open problem since at least Hildebrand's 1989 paper asks whether the Möbius function $\mu$ — the arithmetic function recording the parity of the number of prime factors of an integer — cancels when summed over primes with a fixed shift. This paper claims to settle that conjecture for shift $2$: as $x\to\infty$, $\sum_{p\le x}\mu(p+2)=o(\pi(x))$, equivalently $\sum_{n\le x}\Lambda(n)\mu(n+2)=o(x)$. Earlier results reached the same family of statements only on average over the shift or conditionally on unproved hypotheses such as Siegel zeros. From the cancellation the paper derives a concrete consequence: among primes $p$ with $p+2$ squarefree, the two possible values $\mu(p+2)=1$ and $\mu(p+2)=-1$ occur with equal asymptotic density $1/2$. The conjecture is a well-known model case for the parity problem in sieve theory, so a proof here tests how far current techniques go toward prime-pattern statements without parity-breaking input.

What carries the argument

The argument applies a Vaughan decomposition of the von Mangoldt function $\Lambda$, separating the shifted-Möbius sum into a Type I part, handled by a Bombieri–Vinogradov theorem for $\mu$, and a Type II part. The Type II estimate is the core: a Cauchy–Schwarz step turns it into a determinant square, and the decisive object is Lemma A.1, a root-counting estimate asserting that primitive lattice points with prescribed projective lines modulo a squarefree $S$ are uniformly distributed with density $1/(3\zeta(2))$ and a factor $1/(p+1)$ per prescribed line, with error $O(d_0YMN\,S^2\min\{d_0YM,N\}^{-1/100})$. Around that estimate the paper builds operators on prime-list states — a primitive vector together with ordered tuples of primes dividing its first coordinate — and a parity-conjugacy map $T(u,v)=(u,(u-v)/2)$ that rewrites the shift coefficient $-2$ as $-1$ while preserving the state measure. The root-averaging formula (Proposition 4.4) converts the arithmetic moment into a uniform bound on these operators, and auxiliary weights built from primes in short intervals, averaged over many disjoint prime sets (Proposition 2.3), turn the arbitrary logarithmic savings into the unweighted $o(\pi(x))$ conclusion.

Extended reading notes

Core claim

The central claim is Theorem 1: as $x\to\infty$, $\sum_{p\le x}\mu(p+2)=o(\pi(x))$. Since $\mu$ is $\pm1$ on squarefree integers and $0$ otherwise, the theorem says the $+1$ and $-1$ contributions among shifted primes cancel to lower order than the total number of primes. The paper then proves the companion squarefree density $\sum_{n\le x}\Lambda(n)\mu^2(n+2)=A_2x+O(x/\log^A x)$ for every fixed $A$, with $A_2=\prod_{p>2}(1-1/(p(p-1)))>0$, and combines the two to obtain Theorem 2: for each $\kappa\in\{-1,1\}$, $\#\{p\le x:\mu(p+2)=\kappa\}=(A_2/2+o(1))\pi(x)$. In particular, among primes $p$ with $p+2$ squarefree, the values $1$ and $-1$ each occur with asymptotic density $1/2$.

Load-bearing premise

The whole proof leans on one root-counting estimate, Lemma A.1 in the appendix — adapted from an unpublished preprint and, per the paper's acknowledgements, established with the help of an AI assistant — which asserts a uniform count of primitive lattice points with prescribed projective lines and a specific error term; if that uniformity fails at the claimed precision, the Type II estimate and with it the main theorem do not follow.

Editorial extensions

If this is right

The shift-$2$ case of the Hardy–Littlewood–Chowla conjecture is settled unconditionally: $\mu$ and $\Lambda$ are asymptotically orthogonal, $\sum_{n\le x}\Lambda(n)\mu(n+2)=o(x)$.
The signs of $\mu$ on squarefree shifted primes are balanced: the counts of primes with $\mu(p+2)=1$ and with $\mu(p+2)=-1$ are each $(A_2/2+o(1))\pi(x)$, so each sign has asymptotic density $1/2$ among primes with $\mu^2(p+2)=1$.
The companion density result (Proposition 6.1) holds for every fixed nonzero shift $h$: $\sum_{n\le x}\Lambda(n)\mu^2(n+h)=A_hx+O(x/\log^A x)$ with $A_h=\prod_{p\nmid h}(1-1/(p(p-1)))$, so squarefree shifted primes are equidistributed for every shift.
The paper notes that its Möbius-function discussion has a counterpart for the Liouville function $\lambda$; if that transfer applies here, the same cancellation and equal sign split hold for $\lambda(p+2)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the parity-conjugacy map $T(u,v)=(u,(u-v)/2)$ encodes the specific coefficient $-2$, extending the theorem to other fixed shifts such as $h=4$ probably needs an analogous conjugacy and is the natural stress test of the method.
The sign-splitting conclusion is exactly what the heuristic that $\mu$ on shifted primes behaves like a fair coin predicts, and since the squarefree-density estimate already holds for every fixed shift, sign equidistribution for all shifts is reduced to the analogue of Theorem 1.
Given the provenance the acknowledgements disclose — the appendix's root-counting lemma is adapted from an unpublished preprint and was established with the help of an AI assistant — a direct verification of the uniform error term in (A.1), or of the Kloosterman-type bound (A.2) it rests on for composite and even moduli, is the first check a reader should perform before relying on the main theorem

Original abstract

An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the M\"obius function exhibits cancellation on shifted primes; that is, $$\sum_{p\le x}\mu(p+2)=o(\pi(x)),\quad \text{or equivalently}\quad \sum_{n\le x}\Lambda(n)\mu(n+2)=o(x).$$ This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the two sets $\{p:\, \mu(p+2)=1\}$ and $\{p:\, \mu(p+2)=-1\}$ has asymptotic density $1/2$ among the primes $p$ with $\mu^2(p+2)=1$: \begin{equation*} \lim_{x\to\infty} \frac{\#\{p\le x:\ \mu(p+2)=\kappa\}} {\#\{p\le x:\ \mu^2(p+2)=1\}} =\frac12, \quad \kappa\in\{-1,\,1\}. \end{equation*}

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-15: 1 request, 3,434 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.17 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the M\"obius function exhibits cancellation on shifted primes; that is, $$\sum_{p\le x}\mu(p+2)=o(\pi(x)),\quad \text{or equivalently}\quad \sum_{n\le x}\Lambda(n)\mu(n+2)=o(x).$$ This is also a well-known model case for the parity problem in sieve theory. In this paper, we prove this conjecture. As an application, we show that each of the two sets $\{p:\, \mu(p+2)=1\}$ and $\{p:\, \mu(p+2)=-1\}$ has asymptotic density $1/2$ among the primes $p$ with $\mu^2(p+2)=1$: \begin{equation*} \lim_{x\to\infty} \frac{\#\{p\le x:\ \mu(p+2)=\kappa\}} {\#\{p\le x:\ \mu^2(p+2)=1\}} =\frac12, \quad \kappa\in\{-1,\,1\}. \end{equation*}

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An old conjecture dating back at least to Hildebrand's work in 1989, asserts that the M\"obius function exhibits cancellation on shifted primes; that is, $$\sum_{p\le x}\mu(p+2)=o(\pi(x)),\quad \text{or equivalently}\quad \sum_{n\le x}\Lambda(n)\mu(n+2)=o(x).$$ This is also a well-known model case for the parity problem in sieve theory.	background 1.00	0.03	0.03	asserted 0.61
s1	In this paper, we prove this conjecture.	contribution 0.64	0.96	0.97	asserted 1.00
s2	As an application, we show that each of the two sets $\{p:\, \mu(p+2)=1\}$ and $\{p:\, \mu(p+2)=-1\}$ has asymptotic density $1/2$ among the primes $p$ with $\mu^2(p+2)=1$: \begin{equation} \lim_{x\to\infty} \frac{\#\{p\le x:\ \mu(p+2)=\kappa\}} {\#\{p\le x:\ \mu^2(p+2)=1\}} =\frac12, \quad \kappa\in\{-1,\,1\}. \end{equation}	result 0.97	0.94	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (14)

[11]

OpenAI,The Poisson–Dirichlet law for prime predecessors, OpenAI Math Release preprint, 2026, September 24, 2026 version, 74 pp

View on Pith 2026
[1]

AIM Problem Lists,Sarnak’s conjecture,https://aimpl.org/sarnakconjecture/, Section 5, Problem 5.2

View on Pith
[2]

S. Chowla,The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, vol. 4, Gordon and Breach Science Publishers, 1965

View on Pith 1965
[3]

A. Hildebrand,Additive and multiplicative functions on shifted primes, Proc. London Math. Soc. (3)59 (1989), no. 2, 209–232

View on Pith 1989
[4]

D. Koukoulopoulos,The distribution of prime numbers, Grad. Stud. Math., vol. 203, American Mathematical Society, Providence, RI, 2019

View on Pith 2019
[5]

J. D. Lichtman,Averages of the Möbius function on shifted primes, Q. J. Math.73(2022), no. 2, 729–757

View on Pith 2022
[6]

J. D. Lichtman and J. Teräväinen,On the Hardy–Littlewood–Chowla conjecture on average, Forum Math. Sigma10(2022), Paper No. e57, 17 pp

View on Pith 2022
[7]

K. Matomäki, X. Shao, T. Tao, and J. Teräväinen,Higher uniformity of arithmetic functions in short intervals I. All intervals, Forum Math. Pi11(2023), Paper No. e29

View on Pith 2023
[8]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory. I. Classical theory, Cambridge Stud. Adv. Math., vol. 97, Cambridge University Press, Cambridge, 2007

View on Pith 2007
[9]

M. R. Murty and A. Vatwani,Twin primes and the parity problem, J. Number Theory180(2017), 643–659

View on Pith 2017
[10]

,A remark on a conjecture of chowla, J. Ramanujan Math. Soc.33(2018), no. 2, 111–123

View on Pith 2018
[12]

J. Pintz,An approximation to the twin prime conjecture and the parity phenomenon, Indag. Math. (N.S.)26 (2015), no. 5, 883–896

View on Pith 2015
[13]

O. Robert,On van der Corput’sk-th derivative test for exponential sums, Indag. Math. (N.S.)27(2016), no. 2, 559–589

View on Pith 2016
[14]

T. Tao and J. Teräväinen,The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero, J. Lond. Math. Soc. (2)106(2022), no. 4, 3317–3378. 52 G. LÜ, Z. WANG, AND J. WU (Guangshi Lü) School of Mathematics, Shandong University, Jinan 250100, Shandong, China Email address:gslv@sdu.edu.cn (Zhiwei W ang) School of Mathematics, Shandong University, J...

View on Pith 2022

Forward citations

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math.NT 2026-11 conditional novelty 8.0 of 10

The paper proves that ∑_{p≤x} b(p-1) = κ S x/(log x)^{3/2} + O(x/((log x)^{3/2} sqrt(log log x))), where b is the indicator of sums of two squares, κ is the Landau-Ramanujan constant, and S is an explicit Euler product.
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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [11]

OpenAI,The Poisson–Dirichlet law for prime predecessors, OpenAI Math Release preprint, 2026, September 24, 2026 version, 74 pp

View on Pith 2026

[2] [1]

AIM Problem Lists,Sarnak’s conjecture,https://aimpl.org/sarnakconjecture/, Section 5, Problem 5.2

View on Pith

[3] [2]

S. Chowla,The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, vol. 4, Gordon and Breach Science Publishers, 1965

View on Pith 1965

[4] [3]

A. Hildebrand,Additive and multiplicative functions on shifted primes, Proc. London Math. Soc. (3)59 (1989), no. 2, 209–232

View on Pith 1989

[5] [4]

D. Koukoulopoulos,The distribution of prime numbers, Grad. Stud. Math., vol. 203, American Mathematical Society, Providence, RI, 2019

View on Pith 2019

[6] [5]

J. D. Lichtman,Averages of the Möbius function on shifted primes, Q. J. Math.73(2022), no. 2, 729–757

View on Pith 2022

[7] [6]

J. D. Lichtman and J. Teräväinen,On the Hardy–Littlewood–Chowla conjecture on average, Forum Math. Sigma10(2022), Paper No. e57, 17 pp

View on Pith 2022

[8] [7]

K. Matomäki, X. Shao, T. Tao, and J. Teräväinen,Higher uniformity of arithmetic functions in short intervals I. All intervals, Forum Math. Pi11(2023), Paper No. e29

View on Pith 2023

[9] [8]

H. L. Montgomery and R. C. Vaughan,Multiplicative number theory. I. Classical theory, Cambridge Stud. Adv. Math., vol. 97, Cambridge University Press, Cambridge, 2007

View on Pith 2007

[10] [9]

M. R. Murty and A. Vatwani,Twin primes and the parity problem, J. Number Theory180(2017), 643–659

View on Pith 2017

[11] [10]

,A remark on a conjecture of chowla, J. Ramanujan Math. Soc.33(2018), no. 2, 111–123

View on Pith 2018

[12] [12]

J. Pintz,An approximation to the twin prime conjecture and the parity phenomenon, Indag. Math. (N.S.)26 (2015), no. 5, 883–896

View on Pith 2015

[13] [13]

O. Robert,On van der Corput’sk-th derivative test for exponential sums, Indag. Math. (N.S.)27(2016), no. 2, 559–589

View on Pith 2016

[14] [14]

T. Tao and J. Teräväinen,The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero, J. Lond. Math. Soc. (2)106(2022), no. 4, 3317–3378. 52 G. LÜ, Z. WANG, AND J. WU (Guangshi Lü) School of Mathematics, Shandong University, Jinan 250100, Shandong, China Email address:gslv@sdu.edu.cn (Zhiwei W ang) School of Mathematics, Shandong University, J...

View on Pith 2022