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Correlations of multiplicative functions along deterministic and independent sequences

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Pith's one-line read Correlations of multiplicative functions along deterministic sequences vanish, extending known vanishing results through a transfer identity powered by ergodic disjointness.

desk verdict Extends logarithmic Chowla/Elliott zero results to deterministic totally ergodic compositions, with the heavy lifting delegated to the author's earlier disjointness theorems — a real contribution, honestly limited. read the letter →

arxiv 1908.02732 v2 pith:Y6SNMFGK submitted 2019-08-07 math.NT math.DS

Nikos Frantzikinakis This is my paper · ORCID
classification math.NTmath.DS MSC 11N3737A4511K65
keywords multiplicativefunctionsLiouvillefunctionChowlaconjectureElliottdeterministicsequencesergodictheorylogarithmicaveragesFurstenbergcorrespondence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the conjectured random behaviour of multiplicative functions—above all the Liouville function—persists when the arguments are run through a broad class of deterministic sequences, and that correlations across linearly independent shifts vanish or factor as expected. The headline theorem (Theorem 1.1) says that if the deterministic sequence is totally ergodic (for instance, $a(n)=[n\alpha+\beta]$ with irrational $\alpha>1$) and the product of the multiplicative functions is not weakly pretentious to any Dirichlet character (meaning its logarithmic average distance to every periodic completely multiplicative character remains positive), then the logarithmically averaged correlation $\prod_{j=1}^{\ell} f_j(a(m+n_j))$ converges to $0$. The proof is built on a correlation identity (Theorem 3.1) that expresses the composed correlation as a double average of ordinary correlations; this identity follows from disjointness of the associated measure-preserving systems of multiplicative functions from zero-entropy totally ergodic systems. A parallel set of results (Theorems 1.3, 1.4, 1.5) treats shifts taken from independent or equidistributed families, with the same vanishing or factorization conclusions, using characteristic factors, nilmanifold equidistribution, and short-interval results. Taken together these are the strongest currently known partial realizations of the conjectured vanishing and factorization laws for multiplicative functions along low-complexity sequences and thin shift sets.

What carries the argument

The load-bearing object is the F-system of a bounded sequence: the shift-invariant measure on the space of sequences obtained as a weak-star limit of logarithmic orbit averages. Theorem 3.1 is the key identity: it equates the logarithmic correlation of the composed sequences $f_j\circ a$ with an iterated logarithmic correlation of the original $f_j$, and it is derived from disjointness (Theorem 3.3), which forces the relevant joining to be a product measure. A second mechanism, Theorem 4.3, expresses ordinary logarithmically averaged correlations of multiplicative functions as an average over primes in a residue class of integrals of shifted products; this is what reduces the independent-shift theorems to purely ergodic statements (Propositions 4.4 and 4.5), which are proved by induction on the step of a nilsystem using nilcharacters and an equidistribution lemma for multi-parametric nilorbits (Lemma 4.12).

What would settle it

Compute the logarithmic average of $\lambda([m\alpha])\lambda([(m+1)\alpha])$ for an irrational $\alpha>1$; the theorem predicts the limit is $0$, so a persistent non-zero limit would disprove Theorem 1.1. A more conceptual test: exhibit any bounded multiplicative function whose associated measure-preserving system has non-trivial irrational spectrum (the paper's own candidate is the system $(x,y)\mapsto(x,y+x)$ on $\mathbb{T}^2$), which would break the quoted disjointness theorem and, for $a(n)=[n\alpha]$, put $e^{2\pi i\alpha}$ in the spectrum of the composed sequence.

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Extended reading notes

Core claim

The central discovery is a correlation identity: for a deterministic totally ergodic $a$ and multiplicative functions $f_1,\dots,f_\ell$ with values in the unit disc, after passing to a subsequence of intervals $M'$ we have $$$E^{{\log}}$_{m\in M'}\prod_{j=1}^{\ell} f_j(a(m+n_j)) = $E^{{\log}}$_{n\in M'}$E^{{\log}}$_{m\in M'}\prod_{j=1}^{\ell} f_j(m+a(n+n_j)).$$ Once this holds, the known vanishing theorems for ordinary shifts apply directly, because for each fixed $n$ the arguments $a(n+n_j)$ are just fixed points. The identity is powered by Theorem 3.3: every measure-preserving system attached to a bounded collection of multiplicative functions (its F-system) is disjoint from every zero-entropy totally ergodic system, so the joining used to relate the two averages is necessarily the product joining. For independent shifts, the paper proves that correlations with at least one strongly aperiodic factor vanish as $|n|\to\infty$ along independent sets $S$, and that correlations of real-valued multiplicative functions along weakly independent, jointly equidistributed shifts factorize as a product of single-function logarithmic means.

Load-bearing premise

The proof depends on a previously proved theorem, quoted without reproof, saying that the measure-preserving systems associated with bounded multiplicative functions are disjoint from every zero-entropy totally ergodic system; this theorem is currently established only for logarithmic averages, and if it failed in the stated generality the main conclusion would collapse.

Editorial extensions

If this is right

  • For $a(n)=[n\alpha+\beta]$ with irrational $\alpha>1$ and distinct $n_1,n_2$, the logarithmically averaged two-point correlation of the Liouville function along $a$ is $0$: $E^{\log}_{m\in\mathbb{N}}\lambda(a(m+n_1))\lambda(a(m+n_2))=0$.
  • For odd $\ell$, the $\ell$-point correlations of $\lambda$ along every deterministic totally ergodic sequence vanish for distinct shifts, extending the odd-order vanishing results to composed sequences.
  • Sign patterns of a $\{-1,1\}$-valued multiplicative function that is not weakly pretentious to a Dirichlet character appear along deterministic totally ergodic sequences with the expected frequencies: triple patterns have logarithmic density $1/8$, and quadruple patterns have positive lower density.
  • For shift families with different growth rates or rationally independent Beatty frequencies, correlations with at least one strongly aperiodic factor vanish as $|n|\to\infty$, and all $2^{\ell+1}$ sign patterns of the Liouville function occur along the shifts for all but finitely many $n$.
  • For real-valued multiplicative functions along weakly independent, jointly equidistributed shifts, the logarithmic correlation equals the product of the individual logarithmic means, so the functions behave asymptotically like independent random variables under those shifts.

Reading between the lines

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

  • If the disjointness theorem behind Theorem 3.1 were available for Cesàro averages, the same transfer would upgrade Theorem 1.1 from logarithmic to unweighted averages, giving density-type vanishing along deterministic sequences.
  • The paper's Problem 1—that all joint correlations of $f_j\circ a$ coincide with those of $f_j$—would, if true, imply that the partial sums of $f(a(n))$ diverge for every strongly aperiodic multiplicative $f$, and would remove the Sarnak-type obstruction described in Section 1.6.
  • A natural next test is to relax total ergodicity of $a$ to plain ergodicity; the paper's identity fails there, and the expected failure mode is a periodic obstruction, so one might look for counterexamples among deterministic sequences whose associated systems have irrational spectrum.
  • The factorization in Theorem 1.5 suggests that for real-valued multiplicative functions, logarithmic correlations ignore the fine structure of the shifts beyond congruence-equidistribution, a property that fails for complex-valued functions such as $n^{it}$ and $n^{-it}$, and this may be a useful probe for higher-order correlations beyond the currently accessible cases.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies logarithmic correlations of multiplicative functions along deterministic and independent sequences. The main results are: Theorem 1.1 extends the logarithmically averaged Chowla/Elliott-type vanishing results of Tao and Tao-Teräväinen from the identity sequence to compositions with deterministic totally ergodic sequences; Theorems 1.3 and 1.4 establish vanishing of correlations with shifts in sets or sequences that satisfy linear independence conditions; Theorem 1.5 establishes a product formula for real-valued multiplicative functions with weakly independent, jointly equidistributed shifts; and Theorems 1.6–1.8 deduce sign-pattern results. The deterministic results are proved via a reduction (Theorem 3.1) to a disjointness statement (Theorem 3.3) between F-systems of multiplicative functions and zero-entropy totally ergodic systems, with Theorem 3.3 quoted from the author's prior work with Host. The independent-sequence results are proved via ergodic-theoretic statements (Propositions 4.4 and 4.5) using characteristic factors and equidistribution on nilmanifolds.

Significance. Assuming the quoted structural results are correct at the stated generality, the paper contains substantial and interesting new results: it transfers logarithmically averaged Chowla-type vanishing to deterministic totally ergodic shifts and proves general theorems for independent shifts, including sign-pattern results. The new ergodic statements are proved in detail and the reduction structure is elegant. The paper also honestly documents the limitation that analogous statements for f∘a are unknown and face a concrete obstacle (Section 1.6). However, the deterministic part is fully conditional on the imported disjointness theorem, and the independent-sequence part contains a false intermediate theorem (Theorem 4.1). These issues must be resolved before the claims can be accepted.

major comments (2)
  1. [Section 4, Theorem 4.1 and its proof] Theorem 4.1 is false as stated. In the proof of Theorem 4.1 assuming Theorem 1.3, it is claimed that lim_{n→∞, n∈R} |(a_1(n),...,a_ℓ(n))| = ∞ because otherwise a fixed value would occur infinitely often and would contradict the assumption that S = {(a_1(n),...,a_ℓ(n)): n∈R} has independent elements. This is incorrect: the definition of 'independent elements' concerns solutions in S, and if a fixed vector b ∈ N^ℓ occurs for infinitely many n ∈ R, then S simply contains b once, so the condition is not violated. For example, take ℓ = 2, r = 1, R = N, a_1(n) = a_2(n) = 1, f_0 = 1, f_1 = f_2 = λ, and any sequence of intervals M. Then S = {(1,1)} has independent elements (the equation k·n = 0 has at most one solution in S), but the expression in (24) is Elog_{m∈M} λ(m+1)^2 = 1 for every n, so the asserted limit is 1, not 0. The statement of Theorem 4.1 needs a strengthened hypothesis (e.g., that the a_j are independent sequences, or that |(a_1(n),...,a_ℓ(n))| → ∞ along R), and the proof of Theorem 1.4 must be adjusted accordingly.
  2. [Section 3, Theorem 3.3] Theorem 3.3 is the load-bearing step in the proof of Theorem 1.1, but it is quoted without proof and without stating the precise results from [8, Theorem 1.5] and [7, Proposition 3.12] that imply it. Since Theorem 1.1 requires the disjointness statement for joint F-systems of arbitrary collections of U-valued multiplicative functions, the authors should state the imported theorems explicitly and explain how they combine, or provide a proof. In particular, it should be confirmed that [7, Proposition 3.12] applies to joint F-systems of collections of U-valued multiplicative functions, not only to real-valued or single-function systems. This is a verification gap in the current manuscript.
minor comments (5)
  1. [Abstract] The abstract contains a typographical line break: 'taken al ong de-terministic' should read 'taken along deterministic'.
  2. [Section 4, Theorem 4.7] In the statement of Theorem 4.7, the phrase 'the following limit exists in L2(μ)' is followed by an expression that ends with 'dμ', which is misplaced; the limit should be of the average Ep∈Pd ∏_{j=1}^ℓ T^{pj}F_j (without the trailing 'dμ'), or the integral should be written properly.
  3. [Section 1.2, definition of deterministic sequence] The remark defining deterministic sequences for finite-valued sequences using Cesàro averages is somewhat convoluted and could benefit from a clearer statement of the word-complexity condition.
  4. [Section 4, proof of Lemma 4.10] In the proof of Proposition 4.4 for rotations, the assertion that the sequence (pβ_n)_{p∈P_d} is equidistributed on T for irrational β_n is used without a reference; adding a citation for equidistribution of primes in arithmetic progressions would be helpful.
  5. [Section 4.2, proof of Theorem 1.5] In the proof of Theorems 1.3 and 1.5 assuming Propositions 4.4 and 4.5, the argument relies on the fact that strong aperiodicity implies orthogonality to the Kronecker factor via Proposition 4.2(i); it would be clearer to state explicitly that this gives orthogonality to the rational Kronecker factor needed in Proposition 4.4.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core theorem is a reduction to the independently established disjointness theorem of the author and Host [7,8], whose hypotheses do not include the target result.

full rationale

Theorem 1.1 is not derived from an assumption that contains its own conclusion. The proof has three independent components: (1) Theorem 3.1, proved via Proposition 3.2, converts a logarithmic correlation of f_j(a(m+n_j)) into an iterated average of uncomposed correlations f_j(m + a(n+n_j)); (2) Theorem 3.3 supplies the disjointness premise needed by Proposition 3.2 and is quoted verbatim from the author's prior work with Host [8, Theorem 1.5] and [7, Proposition 3.12]; (3) Tao's two-point theorem [23] and Tao–Teräväinen [26] kill the inner uncomposed averages. The central self-citation is real and load-bearing, but it is not circular: the cited theorem is about F-systems of multiplicative functions and zero-entropy totally ergodic systems, and its stated assumptions do not include deterministic-sequence correlation formulas. The paper does not reprove Theorem 3.3, but the citation is parameter-free external support and does not raise the circularity score. Section 1.6 explicitly concedes the unproved analogous statement for f_j∘a and the possible T(x,y)=(x,y+x) obstruction, which actually confirms that the paper is not silently assuming its target. No fitted parameter is renamed as a prediction, no uniqueness theorem is invoked to exclude alternatives, and no definition builds the conclusion into an input. Hence the derivation chain contains no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities, parameters, or heuristics. It relies on standard ergodic theory, structural results from the author's prior papers [7,8], and analytic number theory input [21,22]. The main assumptions are domain assumptions from prior literature. There are no fitted parameters; the paper proves a priori statements.

assumptions (4)
  • domain assumption Structural theorem for F-systems of multiplicative functions: every F-system of bounded multiplicative functions with values in the unit disc has no irrational spectrum, and the systems are disjoint from all zero entropy totally ergodic systems.
    Theorem 3.3 is quoted from [8, Theorem 1.5] and [7, Proposition 3.12], and it is the main load-bearing input for Theorem 1.1. The paper does not reprove it.
  • domain assumption The entropy decrement argument of Tao is valid for logarithmic averages and gives the correlation identities used in Theorem 4.3.
    Theorem 4.3 is quoted from [8, Theorem 3.8] and used to reduce Theorems 1.3 and 1.5 to ergodic propositions. It is based on prior work [23,26].
  • domain assumption The characteristic factor for multiple ergodic averages along primes is the Host-Kra factor of order l, including the A(d,r0)-variant of Gowers uniformity of the modified von Mangoldt function.
    Theorem 4.7 and Proposition 4.13 are used to reduce Propositions 4.4 and 4.5 to nilsystems. These are known results from ergodic theory and [14].
  • domain assumption Matomaki-Radziwill-Tao short interval results imply the ergodic properties in Proposition 4.2: strong aperiodicity implies orthogonality to the Kronecker factor, and real-valued multiplicative functions have constant conditional expectation on the invariant factor.
    Proposition 4.2 is stated with proof sketches referencing [22, Theorem B.1] and [21, Theorem 1]. These are external theorems with published proofs.

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Cite this review

Pith. "Pith review of Correlations of multiplicative functions along deterministic and independent sequences." pith.science (2026). https://pith.science/paper/Y6SNMFGK

@misc{pith2026190802732,
  author       = {Pith},
  title        = {Pith review of: Correlations of multiplicative functions along deterministic and independent sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y6SNMFGK}},
  note         = {Machine review of arXiv:1908.02732}
}
read the original abstract

We study correlations of multiplicative functions taken along deterministic sequences and sequences that satisfy certain linear independence assumptions. The results obtained extend recent results of Tao and Ter\"av\"ainen and results of the author. Our approach is to use tools from ergodic theory in order to effectively exploit feedback from analytic number theory. The results on deterministic sequences crucially use structural properties of measure preserving systems associated with bounded multiplicative functions that were recently obtained by the author and Host. The results on independent sequences depend on multiple ergodic theorems obtained using the theory of characteristic factors and qualitative equidistribution results on nilmanifolds.

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