Discorrelation of the M\"obius function with linear phases in short intervals

Javier Pliego; Mengdi Wang

REVIEW

Discorrelation of the M\"obius function with linear phases in short intervals

Javier Pliego, Mengdi Wang

Number Theory · Submitted 8 Oct 2026

Pith's verdict: Pliego–Wang break the decades-old 3/5 barrier for Möbius discorrelation with linear phases, using a genuinely new Type III method and three-block partitions; the proof is careful and honest, though it leans on two large-value preprints that are not yet independently verified.

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arXiv ID2610.11487 (version 2)

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CitationJavier Pliego, Mengdi Wang. "Discorrelation of the M\"obius function with linear phases in short intervals." arXiv:2610.11487 (2026). https://pith.science/paper/YAWK3PP7

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Editor's note

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Mengdi and Javier have done something real. The 3/5 barrier for μ(n)e(nα) short-interval sums has held since Zhan (1991) and Matomäki–Teräväinen (2023). They break it with an exponent of 0.58+ε, which is a genuine improvement, even if it looks incremental on paper.

What is new is the Type III analysis—splitting Heath–Brown factors into three blocks instead of the usual two, and using large-value estimates for character-twisted Dirichlet polynomials to handle the (1/5,1/5,1/5) configuration that had resisted earlier approaches. That is a clever structural insight, and they work through the four combinatorial cases (I, I², II, III) with honest casework. The paper is written clearly and acknowledges its own limitations: Section 1.1 lays out exactly why the old methods stalled, and the “Limiting cases” paragraph states that the 0.58 exponent is a hard floor without new arithmetic input.

The soft spot is the same one flagged by the reader: the estimates in Sections 2 and 8 depend on Guth–Maynard (2026) and Chen–Gupta–Li (2026), both still preprints. The arguments internal to this paper are consistent given those external results, but if either preprint contains a bug in the constants, the admissible exponent here might shift or fail. The authors flag this dependence honestly, and this is standard practice in the subfield, but it is a real load-bearing dependence.

A smaller note: the 0.58 exponent is not 0.58+ε in the rigorous statement, and the improvement over 0.6 is modest in absolute terms; still, for a problem stuck for thirty years, movement is significant. Overall: this is a serious paper, carefully argued, with a genuine new idea. It deserves a strong referee despite the dependence on recent preprints. I would send it to peer review. The specialist reader will value the Type III framework even if the final exponent later improves. For my own work I probably won't cite it (different area), but I would mention it in conversation.

Referee report

Minor objections (6)

[§2.1, Lemma 2.2] The statement of Lemma 2.2 requires the hypothesis T′ ≥ T^{1/3}, but this condition is only mentioned in the proof. Please add it explicitly to the statement of the lemma for clarity.
[§6, Lemma 6.1] In the statement of Lemma 6.1, the condition reads M₁, M₂ ≪ H x^{-ε^2}. The exponent ε^2 seems unusual; please check whether this should be x^{-ε/2} or a similar expression, and correct if necessary.
[§8, after (8.9)] The definitions of f_M(σ) and h_M(σ) in (8.9) are clear, but the reader may benefit from a brief sentence explaining why the threshold σ=7/10 and σ=8/10 appear (they correspond to the 7/10-line mentioned later).
[§5.2, Lemma 5.3] The condition on β in the hypothesis is given as x/H² ≤ β ≤ x^ε, but in the application of Proposition 5.4 the upper bound is (log x)^{-2A}. Please ensure consistency; a note that the exponent ε can be replaced by a sufficiently small constant would help.
[§1, Proposition 3.1] The major-arc analysis assumes H ≥ x^{17/30+ε}, while Theorem 1.1 only requires H ≥ x^{0.58+ε}. Since 17/30 ≈ 0.5667, the condition is slightly weaker, so there is no mismatch. However, adding a remark that the major arcs are handled with a weaker exponent would avoid confusion.
[§9, (9.3)] In condition (9.3), the right-hand side reads X_{i∈I} γ_i < 10θ – 16/3. For θ just above 0.58, 10θ – 16/3 ≈ 5.8 – 5.333 = 0.467, which is plausible. Please verify that the constant 16/3 is correct and not a typo for 16/3 of something else.

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The paper

The paper claims

This paper proves that the Möbius function and the von Mangoldt function are decorrelated from linear phases on intervals as short as x^{0.58+ε}, breaking the previous 3/5 barrier.

The reading

The paper proves that for intervals of length H ≥ $x^{{0.58+ε}}$, the exponential sums of the Möbius function and the von Mangoldt function with linear phases are significantly smaller than the trivial bound H, uniformly in the frequency α. For the Möbius function the bound is ≪ H (log x)^{-1/3+ε}; for the von Mangoldt function it is ≪ H (log x)^{-A} for any A unless α is very close to a rational with small denominator. This breaks the longstanding 3/5 barrier (H ≥ $x^{{3/5+ε}}$) from earlier work of Zhan and of Matomäki–Teräväinen. The key innovation is a new Type III estimate that handles configurations in the Heath-Brown decomposition that previously blocked progress. The method is flexible: any future improvement in large value estimates for character-twisted Dirichlet polynomials would immediately allow even shorter intervals.

What carries the argument

The central machinery is a Type III estimate for mean values of products of three Dirichlet polynomials twisted by Dirichlet characters. The paper partitions the Dirichlet polynomials arising from Heath-Brown's identity into three blocks of roughly equal length $x^{{1/5}}$, then applies refined large value estimates (relying on Guth–Maynard 2026 and Chen–Gupta–Li 2026) to bound the number of well-spaced points where all three polynomials have large size. This is combined with stationary-phase arguments that handle two regimes of β (small and large) via different Dirichlet approximations, and with the earlier Type I, Type I2, and Type II analyses that handle other combinatorial configurations.

Extended reading notes

Core claim

The paper claims to have proven that the exponential sum of the Möbius function with linear phases satisfies |∑_{x<n≤x+H} μ(n)e(nα)| ≪ H (log x)^{-1/3+ε} for H ≥ $x^{{0.58+ε}}$, uniformly in α, and that the analogous sum for the von Mangoldt function satisfies |∑_{x<n≤x+H} Λ(n)e(nα)| ≪ H (log x)^{-A} for any A unless ∥qα∥ is very small for some small q. These results improve the previous admissible exponent from 3/5 to 0.58, and the method shows that any further refinement of large value estimates for Dirichlet polynomials will lower the exponent further.

Load-bearing premise

The proofs critically depend on the large value estimates for Dirichlet polynomials established in the preprints [1] (Chen–Gupta–Li 2026) and [3] (Guth–Maynard 2026); if those estimates contain errors or require stronger conditions on q, N, or T, the main results may not hold in the stated range H≥$x^{{0.58+ε}}$.

Editorial extensions

If this is right

The new Type III analysis can likely be adapted to other arithmetic functions, such as divisor functions, to obtain similar discorrelation results in short intervals.
Any future improvement in large value estimates for character-twisted Dirichlet polynomials will automatically lower the exponent 0.58 in Theorem 1.1, as noted by the authors.
Corollary 1.4 gives an asymptotic formula for the number of representations of an odd integer as a sum of three primes in a short interval, conditional on assumption (1.2); this is a concrete application to the ternary Goldbach problem.
The method explicitly identifies a limiting case (five Dirichlet polynomials of length x^{1/5}) that prevents further progress without new arithmetic information, providing a clear target for future work.
The approach unifies and extends earlier Type I, Type I2, and Type II estimates, demonstrating that Type III is the necessary ingredient to go below 3/5.

Reading between the lines

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

The result suggests that the true lower bound for H may be as low as x^{1/2+ε}, since the method's limiting case involves x^{1/5} polynomials, and further improvements in large value estimates could potentially approach that threshold.
The conditional result Theorem 1.3, combined with the discussion on zero-density estimates, highlights a tension between the large value approach and the density approach: the former currently gives better exponents, but the latter might improve if new zero-density estimates in short intervals become available.
The paper's reliance on preprints (Guth–Maynard 2026, Chen–Gupta–Li 2026) means that the unconditional validity of the results depends on the peer-review and potential corrections of those works.
The explicit description of the exceptional set for Λ (the condition on ∥qα∥) links the problem of discorrelation to Diophantine approximation, suggesting a potential connection to metric number theory and equidistribution.

Original abstract

We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequencies over short intervals. As an application, we prove that \[ \Big|\sum_{x<n \leq x+H}\mu(n) e(n\alpha)\Big| \ll H(\log x)^{-1/3+\varepsilon } \] whenever $x^{0.58+\varepsilon }\leq H\leq x$. In the same range of $H$ we also show that \[ \Big|\sum_{x<n \leq x+H} \Lambda (n) e(n\alpha)\Big| \ll H(\log x)^{-A} \] for arbitrarily large $A>0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|q\alpha \| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials on the critical range would lead to a corresponding improvement in the lower bound for $H$.

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, 2026-10-09: 1 request, 5,393 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequency over short intervals. As an application, we prove that \[ \Big|\sum_{x<n \leq x+H}\mu(n) e(n\alpha)\Big| \ll H(\log x)^{-1/3+\varepsilon } \] whenever $x^{0.58+\varepsilon }\leq H\leq x$. In the same range of $H$ we also show that \[ \Big|\sum_{x<n \leq x+H} \Lambda (n) e(n\alpha)\Big| \ll H(\log x)^{-A} \] for arbitrarily large $A>0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\|q\alpha \| \ll x (\log x)^{O_A(1)}/H^2$. This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T. Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272]. Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding improvement in the lower bound for $H$.

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	We introduce a new approach to studying correlations between arithmetic functions and linear phases with respect to highly irrational frequency over short intervals.	contribution 0.91	0.83	0.05	asserted 0.95
s1	As an application, we prove that \[ \Big\|\sum_{x<n \leq x+H}\mu(n) e(n\alpha)\Big\| \ll H(\log x)^{-1/3+\varepsilon } \] whenever $x^{0.58+\varepsilon }\leq H\leq x$.	result 1.00	0.96	0.59	asserted 1.00
s2	In the same range of $H$ we also show that \[ \Big\|\sum_{x<n \leq x+H} \Lambda (n) e(n\alpha)\Big\| \ll H(\log x)^{-A} \] for arbitrarily large $A>0$ unless there exists an integer $1\leq q\ll (\log x)^{O_A(1)}$ such that $\\|q\alpha \\| \ll x (\log x)^{O_A(1)}/H^2$.	result 1.00	0.95	0.17	asserted 1.00
s3	This breaks the $3/5$ barrier in Theorem 1.5 of arXiv:1911.09076v2 and Theorem 2 of [T.	interpretation 0.57	0.83	0.10	asserted 0.99
s4	Zhan, "On the representation of large odd integer as a sum of three almost equal primes," Acta Mathematica Sinica 7.3 (1991), 259-272].	background 0.48	0.08	0.03	unclear 0.75
s5	Moreover, by our method, any improvement in large value estimates for character-twisted Dirichlet polynomials would lead to a corresponding improvement in the lower bound for $H$.	interpretation 0.93	0.77	0.06	asserted 0.74

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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references and citing sentences
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S sections
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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

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