On some classification problems of multiplicative functions

M. Lema\'nczyk; O. Klurman; S. Kasjan

REVIEW 1 major objections 2 minor 34 references

On some classification problems of multiplicative functions

T0 review · 1 major / 2 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that a nonzero multiplicative function is Toeplitz exactly when it agrees with a Dirichlet character off a finite set of primes, and that a pretentious multiplicative function has a unique Furstenberg system exactly when…

desk verdict The main classification theorems are real and the proofs hold up; the one flagged step in Theorem 0.5 is terse but correct. read the letter →

arxiv 2505.07277 v1 pith:PVUGKEY2 submitted 2025-05-12 math.NT math.DS

S. Kasjan , O. Klurman , M. Lemańczyk This is my paper · ORCID

classification math.NTmath.DS MSC 11A2511N3737A45

keywords multiplicativefunctionsToeplitzsequencesDirichletcharacterspretentiousFurstenbergsystemsrationallyalmostperiodicautomaticcorrelationconjecture

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 gives two classification theorems for bounded multiplicative functions. Theorem 0.1 says a nonzero multiplicative function is Toeplitz precisely when, outside a finite set of primes, it equals a Dirichlet character; such functions are necessarily regular, so they determine a unique Furstenberg system. Theorem 0.5 says that among pretentious functions, having exactly one Furstenberg system is equivalent to the twist parameter being zero and the prime series $\sum_p \frac1p(1-f(p)\overline{\chi(p)})$ converging, equivalently to being rationally almost periodic. A corollary is that every real-valued pretentious multiplicative function is generic, and that the corrected correlation conjecture for strongly non-pretentious functions would imply the uniqueness conjecture for Furstenberg systems. The paper also sorts out how automatic, Toeplitz, periodic, and aperiodic classes sit inside the pretentious world.

What carries the argument

The load-bearing objects are the multiplicative distance $D(f,g)^2=\sum_p \frac1p(1-\mathrm{Re}(f(p)\overline{g(p)}))$, Toeplitz period structures with essential periods and $p$-valuations, and Furstenberg systems obtained as weak* limits of shifts of $f$. For Theorem 0.1, multiplicativity forces the set of primes dividing the periods of the position 1 to be finite, so $f$ agrees with a Dirichlet character on all integers coprime to that finite set. For Theorem 0.5, the proof converts uniqueness of the Furstenberg system into convergence of a correlation product $\prod_p M_p$ whose local factors satisfy $M_p=1-\frac{2}{p}(1-f(p)\overline{\chi(p)})+O(p^{-2})$ for large $p$, so convergence of the product is exactly convergence of the displayed prime series; the odometer/discrete-spectrum structure theorem from the theory of Furstenberg systems of pretentious functions supplies the dynamical scaffolding.

What would settle it

To refute Theorem 0.1, exhibit a multiplicative function that is Toeplitz but, for every Dirichlet character and every finite set of primes, differs from the character at some integer coprime to all primes in the set. To refute Theorem 0.5, produce a bounded pretentious $f$ with $D(f,\chi n^{it})<\infty$ whose correlation limits all exist while either $t\neq 0$ or the series $\sum_p \frac1p(1-f(p)\overline{\chi(p)})$ diverges; the paper predicts no such function exists.

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Extended reading notes

Core claim

The central discovery is that two apparently dynamical notions, being Toeplitz and having a unique Furstenberg system, are, for multiplicative functions, purely number-theoretic conditions. Theorem 0.1 states that a nonzero multiplicative $f:\mathbb{N}\to\mathbb{C}$ is Toeplitz iff there exist a Dirichlet character $\chi$ and a finite set $F$ of primes such that $f(n)=\chi(n)$ for every $n$ coprime to every element of $F$, and moreover every such $f$ is regular, hence uniquely ergodic. Theorem 0.5 states that for $f\in\mathcal{M}$ with $D(f,\chi n^{it})<\infty$, the function $f$ is generic (has exactly one Furstenberg system) iff $t=0$ and $\sum_p \frac1p(1-f(p)\overline{\chi(p)})$ converges, iff $f$ is Besicovitch rationally almost periodic, with the unique system being an ergodic odometer. The proof combines the period-structure machinery of Toeplitz sequences with correlation formulas and the structure theorem that Furstenberg systems of pretentious functions are discrete-spectrum odometer systems.

Load-bearing premise

The classification rests on the earlier structure theorem that every Furstenberg system of a pretentious function has discrete spectrum whose ergodic components are odometers; if that theorem were false or incomplete, the equivalence in Theorem 0.5 would lose its foundation.

Editorial extensions

If this is right

Every Toeplitz multiplicative function is regular, so its subshift is uniquely ergodic and its unique Furstenberg system is an odometer.
For pretentious functions, uniqueness of the Furstenberg system is checkable from prime values: $t=0$ and $\sum_p \frac1p(1-f(p)\overline{\chi(p)})<\infty$.
All real-valued pretentious multiplicative functions and all finitely valued pretentious multiplicative functions are rationally almost periodic and generic, with an ergodic odometer as the unique Furstenberg system.
If a pretentious function has $t\neq 0$, it has uncountably many pairwise isomorphic non-ergodic Furstenberg systems; uniqueness and ergodicity always go together.
The corrected correlation conjecture for strongly non-pretentious functions implies the uniqueness conjecture for every real-valued multiplicative function bounded by 1.

Reading between the lines

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

The criterion turns the one-or-uncountably-many dichotomy for Furstenberg systems of pretentious functions into a decision problem: compute $t$ and evaluate the prime series, so uniqueness becomes testable rather than a purely dynamical fact.
The paper's Corollary 2.8 extends the Toeplitz rigidity: even an infinite exceptional set with $\sum_{p\in F}1/p<\infty$ cannot destroy uniqueness; this suggests that any counterexample to uniqueness for non-pretentious functions must be strongly spread out in the multiplicative sense.
One testable extension: if the rate of convergence in the prime series is controlled, one should be able to bound the rate at which correlations of $f$ converge; the paper does not quantify this.
The equivalence between Toeplitz and 'Dirichlet character off a finite set' also explains why the BBC condition is closed under multiplication, whereas automaticity is not; the two notions diverge exactly by the finite exceptional primes.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Core: Theorem 0.1 and Theorem 0.5 are the new substance. The Toeplitz characterization is a clean, unconditional statement that genuinely improves on the automatic-sequence analogy, and the uniqueness criterion for Furstenberg systems gives the field what [13] left open: a number-theoretic way to see genericity. The conditional Corollary 0.6 is honestly labeled and does what it claims. The examples and the aperiodic-class clarifications are useful.

The paper does well to keep the proofs detailed. The structure arguments in Section 1 (Lemma 1.2, Lemma 1.9, Theorem 1.11) are careful and fill a gap that earlier survey treatments left informal. The use of [13]'s structure theorem is legitimate: it is a published recent result, and the paper does not try to reprove it; the new criterion is a separate, genuinely new result.

Now the stress-test. The flagged step in Theorem 0.5 (i)⇒(ii) — the claim that mean existence plus pretentiousness forces f^{\phi(q)}(2^k) = -2^{ki\phi(q)t} — is terse but correct. Halász's theorem says the mean can exist only if the singularity at s=1+i\phi(q)t is removable. Since D(f^{\phi(q)}, n^{i\phi(q)t}) < ∞, the Dirichlet series has a pole at that point coming from the primes where the function pretends to the Archimedean character. The only way to cancel it is a zero of a local factor. For any prime p≥3, the local factor at s=1+iu has modulus at least 1/2 (by the triangle inequality), so it cannot vanish. The 2-local factor can vanish, and equality in the triangle inequality forces g(2^k) = -2^{iku} for every k. So the contradiction argument is sound. The paper would be easier to referee if that step cited a specific lemma (or proved it in a sentence), but it is not wrong.

The weakest point is simply the dependence on the heavy machinery of [13]; readers who don't know that paper will have to take a lot on faith. That is standard for this area, and not a defect of the present work. There are minor typos (the abstract's 'M oreover') and a small clash of notation (F the set vs F the function), but nothing that affects the math.

Verdict: this deserves a serious referee. I'd send it out; the main theorems will be citable. My own take is that it's a solid paper, likely publishable in a good journal without major revision.

Referee Report

1 major / 2 minor

Summary. The paper gives a full classification of multiplicative functions that are Toeplitz (Theorem 0.1: they coincide with a Dirichlet character outside a finite set of primes) and a characterization of pretentious multiplicative functions with a unique Furstenberg system (Theorem 0.5: uniqueness holds iff t=0 and the prime series (11) converges, iff the function is rationally almost periodic). It also proves that all Toeplitz multiplicative functions are regular, that real-valued pretentious functions have a unique Furstenberg system, that the corrected Elliott conjecture implies the Frantzikinakis–Host conjecture, and it clarifies several relations between aperiodicity notions.

Significance. If correct, Theorem 0.1 settles the classification of multiplicative Toeplitz sequences, and Theorem 0.5 provides a clean dichotomy for Furstenberg systems of pretentious functions: one ergodic odometer versus uncountably many isomorphic non-ergodic systems. The combination of pretentious number theory and ergodic theory is elegant, and the paper gives detailed proofs for most steps. The main caveat is a missing justification of a critical pointwise identity in the proof of Theorem 0.5; once that is supplied, the central claims are likely sound.

major comments (1)

[Section 2.2 (proof of Theorem 0.5, (i)⇒(ii))] The step "Since the mean of f^{φ(q)} must exist ... f^{φ(q)}(2^k) = -2^{kiφ(q)t} for all k≥1 in view of the Halász theorem" is not justified. Halász's theorem provides asymptotic formulas for the partial sums of a multiplicative function in terms of the minimum of D(g, n^{it}); it does not directly yield pointwise identities for f(2^k). What is needed is a lemma: if g∈M, D(g, n^{iu})<∞ with u≠0, and the Cesàro mean of g exists, then the Dirichlet series of g must be holomorphic at s=1+iu, which forces the local factor at the prime 2 to satisfy G_2(1+iu)=0; since G_2(z)=∑_{k≥0} g(2^k) z^k has |z|=1/2 and coefficients bounded by 1, the only way this can happen is g(2^k)=-2^{iku} for all k≥1. This argument is absent, and without it the exclusion of t≠0 in (i)⇒(ii) is unsupported. Please either prove this lemma or provide a precise reference stating it.

minor comments (2)

[Proof of Corollary 0.6, Case 2] The sentence "f^2 is pretentious (in fact, it is RAP, see Remark 2.12 in [5])" is inaccurate: for an aperiodic real-valued f, f^2 need not be pretentious (for example, if f(p)=0 on a set of primes with divergent reciprocal sum, then f^2 is not pretentious), even though it is RAP. Since the subsequent argument only uses the RAP property, the claim should be rephrased to state that f^2 is RAP and hence generic.
[Throughout] There are numerous typographical and spacing issues in the abstract and text (e.g., "M oreover" in the abstract, "pretencious" in the introduction). A careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classification theorems are proved from elementary lemmas, external analytic results, and structure theorems that do not contain the target uniqueness criterion.

full rationale

The derivation chain is not circular. Theorem 0.1's necessity direction is proved in Section 1 through Lemmas 1.2, 1.5–1.9 and Theorem 1.11, which construct the Dirichlet character from the periods of the position 1 and use only multiplicativity, Dirichlet's theorem, and the elementary Lemma 1.2; no parameter is fitted and no step assumes the conclusion (6). The sufficiency direction is the explicit computation in footnote 10. Theorem 0.5 is assembled from external and prior results, but none of those results states the target dichotomy. (iii)↔(ii) combines Daboussi–Delange Theorem 6 [7] (external) with Corollary 2.11 of [5] (RAP ⇔ Besicovitch almost periodicity for functions in M); the latter is a structural result about almost periodic multiplicative functions, not about uniqueness of Furstenberg systems. (iii)→(i) uses Theorem 1.7 of [4] that RAP functions are generic. In (i)→(ii), [13] supplies only that spectral measures of pretentious Furstenberg systems are purely atomic, and [25] supplies a correlation expansion; neither asserts that a unique system forces convergence of (11), which is the new content. The paper explicitly labels Corollary 0.6 as conditional on the corrected Elliott conjecture, so no hidden assumption is disguised as a prediction. The one genuinely questionable internal step is the appeal to 'the Halász theorem' at equation (17): it cites no theorem number and, as written, the displayed identity does not follow from standard Halász asymptotics. That is a correctness or proof-completeness concern, not a circularity: the step does not define the conclusion into the hypothesis. Accordingly there are no circular steps and the score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

There are no fitted parameters or invented entities. All unconditional results rest on standard theorems (Dirichlet, PNT in APs, Halasz) and on cited structural results from the same research program. The corrected Elliott conjecture appears only as the hypothesis of a conditional statement, so it does not burden the unconditional theorems.

assumptions (7)

standard math Dirichlet theorem on primes in arithmetic progressions
Invoked in Lemmas 1.5 and 1.6, Remark 1.12 and Proposition 5.1 to select primes in specific residue classes with coprimality conditions.
standard math Prime Number Theorem in arithmetic progressions
Used in the argument establishing (5) that Toeplitz multiplicative functions are pretentious, and in Proposition 4.3 to count primes in dyadic-like intervals.
standard math Halasz theorem on means of multiplicative functions
Used in Corollaries 2.5 and 2.7, Lemma 2.6 and Proposition 3.2 to conclude the Archimedean parameter t vanishes from divergence of a pretentious distance.
domain assumption Furstenberg system structure for pretentious functions (Frantzikinakis-Lemanczyk-de la Rue [13])
Provides discrete spectrum, odometer ergodic components, and the one-or-uncountably-many dichotomy for Furstenberg systems of pretentious functions; foundational for Theorem 0.5, Corollaries 2.2-2.7 and Proposition 3.1.
domain assumption Correlation asymptotic formula of Klurman (Theorem 1.5 in [25])
Gives the Euler-product expression for second-order correlations used in the (i) implies (ii) direction of Theorem 0.5.
domain assumption RAP and Besicovitch almost periodicity characterizations ([5, Cor 2.11 and Remark 2.12], [7, Theorem 6])
Used for the equivalence (iii) iff (ii) in Theorem 0.5 and for the claim that f^2 is RAP in Corollary 0.6.
domain assumption Corrected Elliott conjecture (Matomaki-Radziwill-Tao [32] and Klurman-Mangerel-Teravainen [27])
Assumed in Corollary 0.6 to derive the Frantzikinakis-Host conjecture; the implication is conditional, so no unconditional claim rests on it.

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Pith. "Pith review of On some classification problems of multiplicative functions." pith.science (2026). https://pith.science/paper/PVUGKEY2

@misc{pith2026250507277,
  author       = {Pith},
  title        = {Pith review of: On some classification problems of multiplicative functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVUGKEY2}},
  note         = {Machine review of arXiv:2505.07277}
}

abstract

We prove that a multiplicative function $f:\mathbb{N}\to\mathbb{C}$ is Toeplitz if and only if there are a Dirichlet character $\chi$ and a finite subset $F$ of prime numbers such that $f(n)=\chi(n)$ for each $n$ which is coprime to all numbers from $F$. All such functions bounded by~1 are necessarily pretentious and they have exactly one Furstenberg system. Moreover, we characterize the class of pretentious functions that have precisely one Furstenberg system as those being Besicovitch (rationally) almost periodic. As a consequence, we show that the corrected Elliott's conjecture implies Frantzikinakis-Host's conjecture on the uniqueness of Furstenberg system for all real-valued bounded by~1 multiplicative functions. We also clarify relations between different classes of aperiodic multiplicative functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 32 canonical work pages

[13]

Furstenberg systems of pretentious and MRT multiplicative functions

N. Frantzikinakis, M. Lemańczyk, T. de la Rue, Furstenberg sys- tems of pretentious and MRT multiplicative functions , published on- line 03 February 2025, pp. 1-80, in Ergodic Theory Dynam. Sys tems, arXiv:2304.03121

work page Pith review arXiv 2025
[1]

El Abdalaoui, J

H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk, T. de l a Rue, Möbius disjointness for models of an ergodic system and beyo nd, Israel J. Math. 228 (2018), 707-751

work page 2018
[2]

Allouche, J

J.-P. Allouche, J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge: Cambridge University Press, 2003

work page 2003
[3]

J.P. Bell, N. Bruin, M. Coons, Transcendence of generating func- tions whose coeﬃcients are multiplicative , Transactions of the American Mathematical Society, 364 (2012), 933–959

work page 2012
[4]

Bergelson, J

V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, F. Richt er, Rationally almost periodic sequences, polynomial multiple recurrenc e and symbolic dynamics, Ergodic Theory Dynam. Systems 39 (2019), 2332-2383

work page 2019
[5]

Bergelson, J

V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, F. Richt er, A struc- ture theorem for level sets of multiplicative functions and applications, International Math. Research Notices 2020, no 5, 1300-1345

work page 2020
[6]

Chowla, The Riemann hypothesis and Hilbert’s tenth problem , Math- ematics and its Applications, Vol

S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem , Math- ematics and its Applications, Vol. 4, Gordon and Breach Scie nce Pub- lishers, New York, 1965

work page 1965
[7]

Daboussi, H

H. Daboussi, H. Delange, On multiplicative arithmetical functions whose modulus does not exceed one , J. London Math. Soc. (2) 26 (1982), 245–264

work page 1982

Show all 34 references

[8]

Davenport, Multiplicative number theory , Graduate Texts in Mathe- matics, 74, Springer, New York, 2000

H. Davenport, Multiplicative number theory , Graduate Texts in Mathe- matics, 74, Springer, New York, 2000

2000
[9]

Edeko, On the isomorphism problem for non-minimal transforma- tions with discrete spectrum , Discrete Continuous Dynam

N. Edeko, On the isomorphism problem for non-minimal transforma- tions with discrete spectrum , Discrete Continuous Dynam. Systems 39 (2019), 6001–6021

2019
[10]

Ferenczi, J

S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk, Sarnak’s Conjecture - what’s new , Ergodic Theory and Dynamical Systems in their Interac- tions with Arithmetics and Combinatorics, Lecture Notes in Mathemat- ics 2213, Springer International Publishing, (2018), 163– 235

2018
[11]

Frantzikinakis, B

N. Frantzikinakis, B. Host, Higher order Fourier analysis of multiplica- tive functions and applications , J. Amer. Math. Soc. 30 (2017), 67–157. 39

2017
[12]

Frantzikinakis, B

N. Frantzikinakis, B. Host, The logarithmic Sarnak conjecture for ergodic weights, Annals Math. (2), 187 (2018), 869–931

2018
[14]

Glazkov, Periodic multiplicative number-theoretic functions , (Rus- sian), Volz

V.V. Glazkov, Periodic multiplicative number-theoretic functions , (Rus- sian), Volz. Mat. Sb. Vyp. 2 (1964), 225–229

1964
[15]

Gomilko, M

A. Gomilko, M. Lemańczyk, T. de la Rue, On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwił ł and Tao , J. Modern Dynamics 17 (2021), 529-555

2021
[16]

Górska, M

M. Górska, M. Lemańczyk, T. de la Rue, On orthogonality to uniquely ergodic systems, to appear in J. d’Analyse Math., arXiv:2404.07907

arXiv
[17]

Granville, K

A. Granville, K. Soundararajan, Large character sums: pretentious char- acters and the P/acute.ts1olya-Vinogradov theorem, J. Amer. Math. Soc., 20 (2007), 357–384

2007
[18]

Granville, K

A. Granville, K. Soundararajan, Multiplicative Number Theory: The pretentious approach. Book , manuscript in preparation
[19]

B. Host, B. Kra, Nilpotent structures in ergodic theory , Mathematical Surveys and Monographs, 236. American Mathematical Societ y, Provi- dence, RI, 2018

2018
[20]

Kaczorowski, A

J. Kaczorowski, A. Perelli, On the structure of the Selberg class, I: 0 ď d ď 1, Acta Math. 182 (1999), 207-241

1999
[21]

Kanigowski, J

A. Kanigowski, J. Kułaga-Przymus, M. Lemańczyk, T. de l a Rue, On arithmetic functions orthogonal to deterministic sequenc es, Advances Math. 428 (2023), 68 pp., https://doi.org/10.1016/j.aim.2023.109 138

2023 doi
[22]

Kanigowski, M

A. Kanigowski, M. Lemańczyk, F. Richter, J. Teräväinen , On the local Fourier uniformity problem for small sets , International Math. Research Notices IMRN 2024, no 15, 11488-11512

2024
[23]

Kanold, Über periodische multiplikative zahlentheoretische Funk - tionen, (German) Math

H.-J. Kanold, Über periodische multiplikative zahlentheoretische Funk - tionen, (German) Math. Ann. 144 (1961), 135–141

1961
[24]

Kanold, Über periodische zahlentheoretische Funktionen, (German) Math

H.-J. Kanold, Über periodische zahlentheoretische Funktionen, (German) Math. Ann. 147 (1962), 269–274. 40

1962
[25]

Klurman, Correlations of multiplicative functions and application s, Compos

O. Klurman, Correlations of multiplicative functions and application s, Compos. Math. 153 (2017), 1622–1657

2017
[26]

Klurman, P

O. Klurman, P. Kurlberg, A note on multiplicative automatic sequences, II. Bull. Lond. Math. Soc. 52 (2020), 185–188

2020
[27]

Klurman, A

O. Klurman, A. Mangerel, J. Teräväinen, On Elliott’s conjecture and applications, preprint, arXiv:2304.05344

arXiv
[28]

Koch, Algebraic Number Theory , Berlin, Springer Verlag, 1997

H. Koch, Algebraic Number Theory , Berlin, Springer Verlag, 1997

1997
[29]

Konieczny, On multiplicative automatic sequences , Bull

J. Konieczny, On multiplicative automatic sequences , Bull. London Math. Soc. 52 (2020) 175–184

2020
[30]

Konieczny, M

J. Konieczny, M. Lemańczyk, C. Müllner, Multiplicative automatic se- quences, Math. Z. 300 (2022), 1297-1318

2022
[31]

Leitmann, D

D. Leitmann, D. Wolke, Periodische und multiplikative zahlentheoretis- che Funktionen, (German) Monatsh. Math. 81 (1976), 279–289

1976
[32]

Matomäki, M

K. Matomäki, M. Radziwiłł, T. Tao, An averaged form of Chowla’s conjecture, Algebra & Number Theory 9 (2015), 2167–2196

2015
[33]

Sarnak, Three lectures on the Möbius function randomness and dy- namics, http://publications.ias.edu/sites/default/ﬁles/MobiusFunctionsLectures

P. Sarnak, Three lectures on the Möbius function randomness and dy- namics, http://publications.ias.edu/sites/default/ﬁles/MobiusFunctionsLectures
[34]

S. G. Williams, Toeplitz minimal ﬂows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete 67 (1984), 95-107. Faculty of Mathematics and Computer Science, Nicolaus Cope rnicus Univer- sity, Chopin street 12/18, 87-100 Toruń, Poland skasjan@mat.umk.pl, mlem@mat.umk.pl Scho...

1984

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [13]

Furstenberg systems of pretentious and MRT multiplicative functions

N. Frantzikinakis, M. Lemańczyk, T. de la Rue, Furstenberg sys- tems of pretentious and MRT multiplicative functions , published on- line 03 February 2025, pp. 1-80, in Ergodic Theory Dynam. Sys tems, arXiv:2304.03121

work page Pith review arXiv 2025

[2] [1]

El Abdalaoui, J

H. El Abdalaoui, J. Kułaga-Przymus, M. Lemańczyk, T. de l a Rue, Möbius disjointness for models of an ergodic system and beyo nd, Israel J. Math. 228 (2018), 707-751

work page 2018

[3] [2]

Allouche, J

J.-P. Allouche, J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge: Cambridge University Press, 2003

work page 2003

[4] [3]

J.P. Bell, N. Bruin, M. Coons, Transcendence of generating func- tions whose coeﬃcients are multiplicative , Transactions of the American Mathematical Society, 364 (2012), 933–959

work page 2012

[5] [4]

Bergelson, J

V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, F. Richt er, Rationally almost periodic sequences, polynomial multiple recurrenc e and symbolic dynamics, Ergodic Theory Dynam. Systems 39 (2019), 2332-2383

work page 2019

[6] [5]

Bergelson, J

V. Bergelson, J. Kułaga-Przymus, M. Lemańczyk, F. Richt er, A struc- ture theorem for level sets of multiplicative functions and applications, International Math. Research Notices 2020, no 5, 1300-1345

work page 2020

[7] [6]

Chowla, The Riemann hypothesis and Hilbert’s tenth problem , Math- ematics and its Applications, Vol

S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem , Math- ematics and its Applications, Vol. 4, Gordon and Breach Scie nce Pub- lishers, New York, 1965

work page 1965

[8] [7]

Daboussi, H

H. Daboussi, H. Delange, On multiplicative arithmetical functions whose modulus does not exceed one , J. London Math. Soc. (2) 26 (1982), 245–264

work page 1982