Logarithmic Chowla Correlations Across All Shift Scales

Jizhou Guo

REVIEW

Logarithmic Chowla Correlations Across All Shift Scales

Jizhou Guo

Number Theory · Submitted 24 Aug 2026

Pith's verdict: Genuinely new shift-uniform logarithmic Chowla estimates, carefully engineered; the main risk is unquoted preprint dependencies that need to be pinned before the headline theorems can be certified.

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arXiv ID2608.23500 (version 4)

Pith IDCV3EOZLQ, Pith's permanent record of this paper and its review

CitationJizhou Guo. "Logarithmic Chowla Correlations Across All Shift Scales." arXiv:2608.23500 (2026). https://pith.science/paper/CV3EOZLQ

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Editor's note

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Dear —,

This is a serious paper and the strongest quantitative two-point logarithmic Chowla result so far. The genuinely new thing is the shift quantifier: an absolute power-of-log saving uniformly over every fixed polylogarithmic range (Theorem 1.6), and an almost-all saving over every subpower range exp((log x)^\eta) with an exceptional set of density below every fixed power of log x (Theorem 1.1). The moment results in Theorems 1.2–1.4 are also new in their maximal and tail forms. I spot-checked the parameter arithmetic in Sections 5–7 and 11, and the internal logic is consistent; the paper is unusually careful about hypothesis verification and includes a full parameter ledger.

The soft spots are exactly where the reader's report puts them: the proof leans heavily on unpublished preprints. Proposition 3.2 replays Menon's minor- and major-arc machinery, and Input 2.4 (the centred decoupling) is an asserted specialization of Tao–Teräväinen [10, Thm 3.3] rather than a quoted statement. The paper says the theorem permits arbitrary intervals, arbitrary residues, and distinct bounded functions, but I cannot verify that from the text. Since equations (104) and (196) depend on exactly that scope, I agree that the strongest conclusions are conditional on the scope claim holding. This is an external-dependency risk, not an internal contradiction. The paper does disclose these dependencies in Appendix A, which is honest, but disclosure doesn't fix verifiability.

The central argument holds up as far as I can check. The minor-arc fourth-moment sieve and the endpoint–residue decoupling (Lemma 8.2) are clean, and the one-conductor Landau–Page organisation in Section 9 is plausible. The ineffectivity is inherited from Siegel, not a new flaw. I did not find a load-bearing internal error; the concern is that the tower of external inputs is too high to certify from this manuscript alone.

Who is this for? Analytic number theorists working on Chowla-type correlations and multiplicative functions in short intervals. It deserves a serious referee: the novelty is real, and the preprint dependencies can be resolved by quoting the exact statements or waiting for Menon's and Tao–Teräväinen's papers to appear. My recommendation is to send it out, but insist that the author identify precisely which versions of the external theorems are being used and quote the exact statements at the interfaces, particularly Input 2.4 and Proposition 3.2.

Best,

Referee report

Significance

If the main theorems are correct, this is a substantial quantitative advance: Theorem 1.1 reaches every fixed stretched-logarithmic subpower shift range with a saving exponent and an explicitly sparse exceptional set, and Theorem 1.6 removes shift exceptions entirely at fixed polylogarithmic ranges with an absolute saving exponent. The moment theorems in Theorem 1.2 and Theorem 1.3 are also strong, with no Landau-Page alternative in Theorem 1.3, and the paper is unusually careful about parameter dependencies, hypothesis verification, and the order of limits: the appendix parameter ledger and the explicit discharge of the hypotheses of Proposition 3.2 are valuable. No internal inconsistency or circularity was found; the external inputs are not self-citations, and the conditional structure is clearly flagged. The main weakness is that the two central inputs, the localized Menon material and the exact scope of the Tao--Teräväinen centered decoupling theorem, are not quoted or made fully checkable from the present text.

Major objections (3)

[§2.3, Input 2.4; Eqs. (104) and (196)] The exact scope of Tao--Teräväinen [10, Theorem 3.3] is load-bearing. The paper asserts that the theorem permits an arbitrary integer interval of length L≥exp(R^{2.01}), arbitrary residue assignments b_p mod p, and distinct 1-bounded functions g_1,g_2, and it explicitly says that no translation-invariance extension is assumed. But the theorem is not quoted, and equations (104) and (196) depend precisely on the arbitrary-interval and arbitrary-residue form. Please include a precise statement or quotation of [10, Theorem 3.3] and verify that the hypotheses hold at (103) and (197), including the second block system in Lemma 10.1. If the published theorem only covers intervals anchored at 1 or of the form [N,2N], or if it requires residues tied to the shift in a special way, then the bounds (105) and (196) do not follow and Theorems 1.6 and 1.1 lose their stated range.
[§3--§4, Proposition 3.2 and Input 2.3] The proof of the central flexible Menon estimate is not self-contained. Section 4 applies Input 2.3, which is literally Corollary 3.7 of the unpublished Menon preprint arXiv:2607.15574v1, and Lemma 3.3 replays minor-arc machinery from the same source. The text claims that no step of Menon's argument uses his printed numerical upper bound on Ω, but this audit cannot be checked from the present paper because the relevant statements of Menon, including his Proposition 3.1 and Corollary 3.7, are not reproduced. Please provide precise statements of these inputs and either prove the needed localizations directly or specify exactly which statements in the Menon preprint are being used. Since Proposition 3.2 is the engine behind Theorems 1.6 and 1.1, this dependency must be made checkable.
[§9.3, Lemma 9.3] The claim that one Landau--Page conductor controls all KMT calls is load-bearing for the exceptional set in Theorem 1.1. The proof invokes the KMT deletion argument and states that a real character can survive only if its L-function has a real zero in a region "contained, uniformly over (162), in θ>1−C R^{-(γ−200a)}", but the text does not quote the relevant KMT zero-free-region or main-term-deletion statement, nor does it verify that the uniformity extends over the full family of parent intervals and subset-rescaled bases in (161). Please supply the exact KMT statement being used and the deduction of (165). Without this, the density bound (171) and the at-most-one-prime assertion are not fully supported.

Minor objections (3)

[Appendix A and §2--§4] The appendix correctly records that Menon's arXiv:2607.15574v1 and Tao--Teräväinen arXiv:2512.01739v2 are preprints, but this fact should also be stated at the points of first use in Sections 2--4, since the main theorems depend directly on those inputs.
[Lemma 11.4] The discretization step from the real-measure exceptional set in Input 2.5 to integer starts is compressed; in particular, the claim that the transformed interval has length K/d^2+O(1) depending only on the residue s should be written out explicitly, including the two possible integer lengths, so that the number of exceptional starts is unambiguous.
[References [7], [10]] The references to the two arXiv preprints should include the date of the version actually used and should be updated if revised versions appear before publication.

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The paper

The paper claims

Every fixed subpower range h≤exp((log x)^η) has a power-of-log saving for the logarithmically averaged Liouville correlation outside an exception set of size at most exp((log x)^η)(log x)^{-C} containing at most one prime.

The reading

This paper proves quantitative bounds for the logarithmically averaged two-point correlations of the Liouville function, $\sum_{n\le x}\lambda(n)\lambda(n+h)/n$, uniformly as $h$ ranges over much larger sets than one fixed shift. The main theorem says that for every fixed $0<\eta<1$ and every $C>0$, all sufficiently large $x$ admit an exceptional set $\mathcal E_{x,\eta}\subseteq[1,\exp((\log x)^\eta)]$ of size at most $\exp((\log x)^\eta)(\log x)^{-C}$, with at most one prime, outside which the correlation is at most $(\log x)^{1-c_\eta}$. A second theorem gives a saving with an absolute exponent for every $h\le(\log x)^A$, with no exceptions at all. The same paper proves large-window moment bounds: averaged over any $H$ consecutive shifts, the $p$-th power of the maximal correlation is $O_p(1+x/H)$, and the $x/H$ factor can be removed when $H\ge x^\theta$ for $\theta>1/3$. This matters because it turns a conjectured qualitative decay into a quantitative uniform statement over extremely long shift ranges, with the obstruction isolated to a single possible exceptional character.

What carries the argument

The load-bearing object is an arithmetic typical set $T(P_1,Q_1,X_0)$: integers $m\ge1$ that have, in each of a chain of prime intervals $[P_j,Q_j]$, at least one prime divisor. The proof's engine is Proposition 3.2, a localized exponential-sum estimate for short sums of $\lambda(n)e(\alpha n)$ over this typical set, in which the circle cutoff $\Omega$ may be any fixed power of the outer logarithm rather than a fixed exponent. Around rational frequencies, the identity $1_T(dm)\lambda(dm)=\lambda(d)1_T(m)\lambda(m)$ holds when $d$ has no prime factor in the selected ranges, and this allows the residue-class split that reduces the estimate to a short-progression variance estimate; the only possible main term there is organized by one Landau--Page conductor. For the super-polylogarithmic theorem a second new object is the progression-of-starts sum $F_{h,M}(\alpha;Z)=\sum_{Z<y\le2Z}|\sum_{t\le M}\lambda(y+ht)e(\alpha t)|$; a decoupling lemma converts the moving residue class of the endpoint into the full residue-class variance before Cauchy--Schwarz, avoiding a factor $\sqrt{hq}$. The long-shift and moment theorems use a different mechanism: convolution by $\lambda(n)/n$ is a family of uniformly bounded maximal operators on $\ell^p(\mathbb{Z})$.

Extended reading notes

Core claim

The paper's central claim is that the logarithmically averaged two-point Chowla correlation can be controlled uniformly over growing shift ranges with a genuine power-of-log saving. Concretely, Theorem 1.1 asserts that for every fixed $0<\eta<1$ and $C>0$, there is $c_\eta>0$ such that for all sufficiently large $x$ one can remove a set $\mathcal E_{x,\eta}\subseteq[1,\exp((\log x)^\eta)]$ of size $\ll_{\eta,C}\exp((\log x)^\eta)(\log x)^{-C}$, containing at most one prime, and for every remaining $h$, $$\max_{h\le\exp((\log x)^\eta),\,h\notin\mathcal E_{x,\eta}}\left|\sum_{n\le x}\frac{\$\lambda$(n)\$\lambda$(n+h)}{n}\right|\ll_\eta(\log x)^{1-c_\eta}.$$ The exceptional set is generated by a possible Landau--Page real character and is empty when no such character exists, so under the generalized Riemann hypothesis the bound holds for every shift in the range. The paper also proves an absolute-saving version for every fixed polylogarithmic range, all-moment shift-window bounds with a maximal supremum over terminal points, and removal of the $x/H$ factor for windows of length $H\ge x^\theta$ with $\theta>1/3$.

Load-bearing premise

The load-bearing premise is that the unpublished character estimate imported into Proposition 3.2 really can tolerate frequency-cutoff parameters as large as any fixed power of the logarithm, and that the imported centred-decoupling theorem really allows arbitrary intervals and residue classes; if either flexibility is narrower than assumed, the exceptional-set theorems lose their stated range.

Editorial extensions

If this is right

For every fixed $0<\eta<1$, the shift range $h\le\exp((\log x)^\eta)$ has an exceptional set of relative density $(\log x)^{-C}$ with at most one prime; if no Landau--Page character exists, or under GRH, the bound holds for every shift with no exception.
For every fixed $A>0$, all shifts $h\le(\log x)^A$ satisfy $|\sum_{n\le x}\lambda(n)\lambda(n+h)/n|\ll_A(\log x)^{1-c}$ with one absolute $c$, so no exceptional set is needed at fixed polylogarithmic scale.
For $H\ge x^\theta$ with $\theta>1/3$, averaging over any $H$ consecutive shifts makes all fixed $p$-th moments of $\sup_{y\le x}|\sum_{n\le y}\lambda(n)b(n+h)/n|$ uniformly bounded; consequently all but $O(H(\log x)^{-C})$ shifts have maximal terminal correlation at most $(\log x)^\varepsilon$.
For every $H\ge10$, quantitative first- and second-moment bounds hold for the averaged correlation, and if $H(x)\to\infty$ then $\sum_{n\le x}\lambda(n)\lambda(n+h)/n=o(\log x)$ for almost every $h\le H(x)$.
A diagonal argument yields a nonexplicit slowly growing function $\omega(x)\to\infty$ such that the all-shifts power-of-log saving holds for $h\le(\log x)^{\omega(x)}$.

Reading between the lines

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

Beyond the paper: if the imported estimates hold with exactly the stated flexibility, the single Landau--Page conductor is the only obstruction to an all-shifts version of Theorem 1.1; a quantitative zero-free region for that character's $L$-function would remove the exceptional set entirely.
Beyond the paper: the $\theta>1/3$ threshold in Theorem 1.3 is inherited from the almost-all Fourier-uniformity input; any future lowering of that exponent should transfer through the same square-divisor and translated-block argument, yielding all-moment bounds on shorter polynomial windows.
Beyond the paper: the maximal harmonic-multiplier proof is functional-analytic once the uniform exponential-sum estimate is granted; testing the same operator on a random $\pm1$ sequence would reveal which part of the bound is arithmetic cancellation and which is generic convolution structure.
Beyond the paper: because the final normalization is logarithmic, these results do not imply the unweighted Chowla conjecture even for $h=1$; the progression-of-starts Fourier estimate would need a genuinely different major-arc argument to feed natural averages.

Original abstract

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\ll(\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.

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, 2026-09-23: 1 request, 6,733 input tokens.

What the abstract itself states, as probabilities: claims first 0.15 · states limitation 0.61 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.96 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\ll(\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.

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	Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function.	background 1.00	0.05	0.01	asserted 0.88
s1	We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range.	contribution 0.53	0.97	0.79	asserted 1.00
s2	There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $\|\mathcal E_x\cap[1,H]\|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left\|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right\|\ll(\log x)^{1-c}$.	result 0.99	0.94	0.76	asserted 1.00
s3	The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts.	result 0.89	0.90	0.12	asserted 1.00
s4	Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$.	interpretation 0.48	0.75	0.20	asserted 0.99
s5	The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints.	method 1.00	0.90	0.03	asserted 0.95
s6	Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range.	method 0.93	0.79	0.04	asserted 0.95
s7	Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left\|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right\|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.	result 0.92	0.94	0.10	asserted 0.96

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

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sources read

arXiv listing abstract v4

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 3 sentences without a confident label

Uses the method or tool of 2 papers

These are Pilatte's Lemma C.2 and Menon's Proposition 2.13, respectively [ref] .

Improved bounds for multiplicative functions in almost all short intervals · Pilatte's prime blocks · 0.87
Interface source use in this paper Pilatte block geometry and harmonic masses [ref] [ref]–[ref]; separation, normalization, and dilation Pilatte fourth moment [ref] Input [ref] and [ref] Typical-set complement [ref] Input [ref] and [ref] Character-twisted short intervals [ref] Input [ref]; major arcs after the residue-gcd split, with the auxiliary last-range parameter fixed first MRT minor-arc machinery [ref] Lemma [ref] replays the argument and exposes every circle-cutoff dependency Centred arbitrary-interval estimate [ref] Input [ref] and [ref] Hybrid short-progressions variance and good-modulus structure [ref] [ref]; the subpower rational major arc; Sections [ref]–[ref]; and the GRH interface in Section [ref] Davenport Möbius exponential sums [ref] the explicit square-divisor transfer in [ref]; dyadic harmonic multiplier decay and the long-shift moment theorem Almost-all maximal short

Quantitative correlations and some problems on prime factors of consecutive integers · Sources and proof interfaces · 0.95

Compares with 1 paper

On the other hand, the averaged-shift theorem of Matomäki, Radziwiłł, and Tao reaches the full range of shift lengths [ref] ; its later quantitative refinement in [ref] remains an average over the shift.

Improved bounds for multiplicative functions in almost all short intervals · Introduction and main results · 0.82

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