We prove that any $q$-automatic completely multiplicative function
$f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character.
This answers a question of J. P. Allouche and L. Goldmakher and confirms a
conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative
functions. Further, assuming two standard conjectures in number theory, the
methods allows for removing the assumption of completeness

Klurman, Oleksiy

Kurlberg, Pär

English

arXiv

Oleksiy Klurman

Pär Kurlberg

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 357 (2019) 752–755Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comNumber theory/CombinatoricsA note on multiplicative automatic sequences ✩Une note sur les séquences multiplicatives automatiquesOleksiy Klurman, Pär KurlbergDepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, Swedena r t i c l e i n f o a b s t r a c tArticle history:Received 29 April 2019Accepted after revision 7 October 2019Available online 16 October 2019Presented by the Editorial BoardWe prove that any q-automatic completely multiplicative function f : N → C essentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche and L. Gold-makher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely mul-tiplicative functions. Further, assuming GRH, the methods allow us to replace completely multiplicative functions with multiplicative functions.© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous montrons qu’une fonction complètement multiplicative f :N →C qui est également q-automatique est ultimement périodique, coïncidant donc avec un caractère de Dirichlet pour tout nombre premier suffisamment grand. Ceci résout un problème de J.-P. Allouche et L. Goldmakher et confirme une conjecture de J. Bell, N. Bruin et M. Coons pour les fonctions complètement multiplicatives. De plus, sous l’hypothèse de Riemann généralisée, notre démonstration peut être adaptée pour prouver cette conjecture pour toute fonction simplement supposée multiplicative.© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionAutomatic sequences play an important role in computer science and number theory. For a detailed account of the theory and applications, we refer the reader to the classical monograph [2]. One of the applications of such sequences in number theory stems from a celebrated theorem of Christol [4] (also cf. [5]), which asserts that, in order to show the transcendence of the power series ∑n≥1 f (n)zn , it is enough to establish that the function f :N →C is not automatic. In this note, rather than working within the general set up, we confine ourselves to functions taking their values in C. There are several equivalent definitions of automatic (or more precisely, q-automatic) sequences. It will be convenient for us to use the following one.✩ P.K. was partially supported by the Swedish Research Council (2016-03701).E-mail addresses: lklurman@gmail.com (O. Klurman), kurlberg@math.kth.se (P. Kurlberg).https://doi.org/10.1016/j.crma.2019.10.0021631-073X/© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.O. Klurman, P. Kurlberg / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 752–755 753Definition 1.1. The sequence f :N →C is called q-automatic if its q-kernel, defined as a set of subsequencesKq( f ) ={{ f (qin + r}n≥0| i ≥ 1,0 ≤ r ≤ qi − 1}is finite.We remark that any q-automatic sequence takes only finitely many values, since it is a function of the states of a finite automation. A function f : N → C is called completely multiplicative if f (mn) = f (m) f (n) for all m, n ∈ N . The question of which multiplicative functions are q-automatic attracted considerable attention from several authors, including [14], [13], [3], [12], [11] and [1]. In particular, the following conjecture was made in [3].Conjecture 1.2 (Bell–Bruin–Coons). For any multiplicative q-automatic function f : N → C, there exists an eventually peri-odic function g :N →C, such that f (p) = g(p) for all primes p.This conjecture is still open in general, although some progress has been made when f is assumed to be completely multiplicative. In particular, Schlage-Puchta [13] showed that a completely multiplicative q-automatic sequence that does not vanish is almost periodic. Hu [8] improved on that result by showing that the same conclusion holds under a slightly weaker hypothesis. Our first result confirms a strong form of Conjecture 1.2 when f is additionally assumed to be a completelymultiplicative function.Theorem 1.3. Let q ≥ 2 and let f : N → C be a completely multiplicative q-automatic sequence. Then, there exists a Dirichlet char-acter of conductor Q such that either f (n) = χ(n), for all n ∈N such that (n, Q ) = 1 or f (p) = 0 for all sufficiently large p.Theorem 1.3 also confirms the first part of a Conjecture 4.2 in [1]. We remark that a similar result has been very recently obtained independently by Li [11] using combinatorial methods relying on the techniques developed in the theory of automatic sequences. Our proof is shorter and builds upon two deep number theoretic results. Further, assuming the generalized Riemann hypothesis (which in particular implies a strong form of the Artin primitive root conjecture for primes in progressions), our method can be adapted to show the full conjecture (i.e. the assumption on complete multiplicativity can be removed).Note added in July 2019. While the present article was under submission, the authors [9] and independently J. Konieczny [10]found rather different ways to settle Conjecture 1.2 unconditionally. Both approaches use a modification of our Proposi-tion 2.1 to deal with the case where f vanishes on finitely many primes, but then deviate significantly from the approach taken here. In [10], the author proceeds by using the structural theory of automatic sequences, in particular the notion of “arid sets,” to deal with the remaining cases. In our work [9], instead of using the connection to the Artin conjecture (as in the present paper), we establish certain properties of the divisors of the sequences {aqn + b}n≥1, for a, b, q ∈N , to finish up the proof.2. Proof of the main resultWe begin with a simple albeit important remark. Since f is q-automatic and completely multiplicative, the image of f :N →C is finite and therefore, for any prime p, f (p) = 0 or f (p), is a root of unity.Proposition 2.1. Let f :N →C be a q-automatic completely multiplicative function and let M0 = {p| f (p) = 0}. If |M0| < ∞, then there exists a Dirichlet character χ :N →C such that f (p) = χ(p) for all p /∈ M0 .Proof. Since f is q-automatic, there exist positive integers i1 = i2 such that f (qi1n + 1) = f (qi2n + 1) for all n ≥ 1. If n = m ∏p∈M0 p, thenf (qi1m∏p∈M0 p + 1)f (qi2m∏p∈M0 p + 1)= 1 = 0,for all m ≥ 1. The conclusion now immediately follows from Theorem 2 of [6]. Suppose that |M0| = ∞. We are going to show that, in this case, f (p) = 0 for all sufficiently large primes p. Replacing f by | f |, which is also q-automatic, it is enough to prove the claim for the binary valued f :N → {0, 1}.Let 1,n = [1, n] ∩Z. Since f is q-automatic, the q-kernel of f is finite and therefore there exists k0 = k0( f ), such that for all i ≥ 1 and 0 ≤ r ≤ qi − 1, the equalities f (qin + r) = 0 for n ∈ 1,k0 imply f (qin + r) = 0 for all n ≥ 1.754 O. Klurman, P. Kurlberg / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 752–755Lemma 2.2. Suppose that |M0| = ∞. For k0 as above and for any p1, p2, . . . , pk0 ∈ M0 , such that p j > max{q, k0} for all j ∈ 1,k0 , there exists r = r(q, p1, . . . , pk0) such that (r, qpi) = 1 for all i ∈ 1,k0 and f (n ∏i≤k0 pi + r) = 0 for all n ≥ 1. We may further assume that r ≡ 3 (mod 16), and (r − 1, ∏i≤k0 pi) = 1.Proof. For an integer parameter A ≥ logq pk0 , which we shall choose later, by the Chinese remainder theorem, there exists rA such that (rA, q) = 1 and rA ≡ −sq2A (mod ps) for all s ∈ 1,k0. Since ps|q2A s + rA , we have f (q2An + rA) = 0 for all n ∈ 1,k0. The latter implies that f (q2An + rA) = 0 for all n ≥ 1. We claim that f (rA) = 0. Indeed, if this is not the case, we choose a prime p such that f (p) = 1 and consider m = pϕ(q2A)rA , where ϕ(n) denotes the usual Euler totient function. Clearly, m ≡ rA (mod q2A), and consequently 0 = f (m) = ( f (p))ϕ(q2A) f (rA) = 1, which is a contradiction. Note that the same argument works for n ∏i≤k0 pi + rA in place of rA , and therefore we conclude that f (n ∏i≤k0 pi + rA) = 0 for all n ≥ 1. Setting r = rA finishes the proof. Next, without loss of generality, we may assume that there exist three sufficiently large primes t, t′, t′′ > max(q, k0)such that f (t) = f (t′) = f (t′′) = 1, since otherwise we are done. We write ∏a<p≤b p to denote the product of all primes a < p ≤ b. We will require the following consequence of a result due to Heath-Brown [7].Lemma 2.3. Given distinct primes t, t′, t′′ > max(q, k0) and r = r(q, p1, . . . , pk0) as in Lemma 2.2, there exists infinitely many primes qi ≡ r (mod 16 ∏i≤k0 pi) such that at least one of t, t′, t′′ (say t) is a primitive root modulo qi . Moreover, by passing to a subsequence, we may assume that for such primes (qi − 1, q j − 1) = 2 for i = j, and for each l ∈ 1,k0 we have (l/qi) = 1 for all i ∈ 1,k0 .Proof. Let v = 16(∏i≤k0 pi) ∏2<p≤k0 p and chose u such that u ≡ 3 (mod 16) and u ≡ r (mod∏i≤k0 pi), with r as in Lemma 2.2. Moreover, by quadratic reciprocity, we may further select u (mod∏2<p≤k0 p) such that (u/p) = 1 for all primes p ≤ k0, and (u − 1, ∏2<p≤k0 p) = 1. In particular, we have (−3/p) = −1 for any prime p ≡ u mod v . Applying Lemma 3 of [7], with u, v as above and k = 1 (and K = 2k = 2), there exists α ∈ (1/4, 1/2] and δ > 0 such that|{p ≤ x : p ≡ u (mod v), (p − 1)/K = P2(α, δ)|  x/(log x)2,with the implied constant possibly depending on α, with P2(α, δ) denoting the union of the set of primes, together with the set of almost primes n = t1t2 with t1 < t2 both primes, and t1 ∈ [nα, n1/2−δ]. Heath-Brown’s argument then shows that at least one of t, t′, t′′ is a primitive root for infinitely many primes p ≡ u (mod v). Whether the primes qi produced have the properties that (qi − 1)/2 is prime, or that (qi − 1)/2 = t1t2, we may pass to an infinite subsequence of primes q1 < q2 < . . .(satisfying q1 > q) so that (qi − 1, q j − 1) = 2 for i = j (for the latter case of almost primes, note that both t1 and t2 are growing). Proposition 2.4. Suppose that |M0| = ∞. Then, f (p) = 0 for all sufficiently large primes p.Proof. By Lemma 2.3, we may select prime t with f (t) = 1, which is a primitive root modulo infinitely primes q1 <q2 · · · < qk0 (satisfying q1 > max(k0, q)) such that qi ≡ r (mod 16 ∏j≤k0 pk0 ) and consequently f (qi) = 0. From the proof of Lemma 2.2 it follows that there exists rA such that f (n ∏i≤k0 qi + rA) = for all n ≥ 1. Since t is a primitive root modulo q j for j ∈ 1,k0, there exists γ j such that tγ j ≡ rA (mod q j) for j ∈ 1,k0. By the construction and Lemma 2.3, we have (rA/qi) = (−iq2A/qi) = −1 and thus all γi have the same parity. Consequently, by the Chinese remainder theorem, we can choose γ ∈ N , such that γ ≡ γ j (mod q j − 1) for all j ∈ 1,k0. For γ defined this way, we have tγ ≡ rA (mod ∏ j≤k0 q j). Hence, f (tγ ) must be zero. On the other hand, f (tγ ) = f (t)γ = 1, and this contradiction finishes the proof. Combining Proposition 2.1 and Proposition 2.4 yields the conclusion of Theorem 1.3.AcknowledgementsThe first author learnt about the question considered here while attending an excellent talk of J.-P. Allouche at the “Institut Henri-Poincaré” (IHP), Paris. We are also grateful to the referee for valuable comments and suggestions. O.K. would like to thank IHP for excellent working conditions during his visit.References[1] J.-P. Allouche, L. Goldmakher, Mock characters and the Kronecker symbol, J. Number Theory 192 (2018) 356–372.[2] J.-P. Allouche, J. Shallit, Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, UK, 2003.[3] J.P. Bell, N. Bruin, M. Coons, Transcendence of generating functions whose coefficients are multiplicative, Trans. Amer. Math. Soc. 364 (2) (2012) 933–959.[4] G. Christol, Ensembles presque periodiques k-reconnaissables, Theor. Comput. Sci. 9 (1) (1979) 141–145.[5] G. Christol, T. Kamae, M. Mendès France, G. Rauzy, Suites algébriques, automates et substitutions, Bull. Soc. Math. Fr. 108 (4) (1980) 401–419.O. Klurman, P. Kurlberg / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 752–755 755[6] P.D.T.A. Elliott, J. Kish, Harmonic analysis on the positive rationals. Determination of the group generated by the ratios (an + b)/(An + B), Mathematika 63 (3) (2017) 919–943.[7] D.R. Heath-Brown, Artin’s conjecture for primitive roots, Q. J. Math. Oxford, Ser. 2 37 (145) (1986) 27–38.[8] Y. Hu, Subword complexity and non-automaticity of certain completely multiplicative functions, Adv. Appl. Math. 84 (2017) 73–81.[9] O. Klurman, P. Kurlberg, A note on multiplicative automatic sequences, II, https://arxiv.org /abs /1905 .10897, 2019.[10] J. Konieczny, On multiplicative automatic sequences, https://arxiv.org /abs /1905 .11981, 2019.[11] S. Li, On completely multiplicative automatic sequences, https://arxiv.org /pdf /1903 .04385 .pdf.[12] J.-C. Schlage-Puchta, A criterion for non-automaticity of sequences, J. Integer Seq. 6 (3) (2003) 03.3.8, 5.[13] J.-C. Schlage-Puchta, Completely multiplicative automatic functions, Integers 11 (8) (2011) A31.[14] S. Yazdani, Multiplicative functions and k-automatic sequences, J. Théor. Nr. Bordx. 13 (2) (2001) 651–658.

A note on multiplicative automatic sequences

Numérisation de Documents Anciens Mathématiques

We prove that any q-automatic completely multiplicative function essentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming GRH, the methods allow us to replace completely multiplicative functions with multiplicative functions

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                          Klurman, O., & Kurlberg, P. (2019). A note on multiplicative automaticsequences. Comptes Rendus Mathematique, 357(10), 752-755.https://doi.org/10.1016/j.crma.2019.10.002Peer reviewed versionLicense (if available):CC BY-NC-NDLink to published version (if available):10.1016/j.crma.2019.10.002Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Elsevier at https://doi.org/10.1016/j.crma.2019.10.002 . Please refer to any applicable terms of use of thepublisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/A NOTE ON MULTIPLICATIVE AUTOMATIC SEQUENCESOLEKSIY KLURMAN AND PÄR KURLBERGRÉSUMÉ. Nous montrons qu’une fonction complètement multiplicative f : N → C qui estégalement q-automatique est ultimement périodique, coı̈ncidant donc avec un caractère de Diri-chlet pour tout nombre premier suffisamment grand. Ceci résout un problème de J.-P. Alloucheet L. Goldmakher, et confirme une conjecture de J. Bell, N. Bruin et M. Coons pour les fonc-tions complètement multiplicatives. De plus, sous l’hypothèse de Riemann généralisée, notredémostration peut être adaptée pour démontrer cette conjecture pour toute fonction simplementsupposée multiplicative.ABSTRACT. We prove that any q-automatic completely multiplicative function f : N → Cessentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche andL. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completelymultiplicative functions. Further, assuming GRH, the methods allow us to replace completelymultiplicative functions with multiplicative functions.1. INTRODUCTIONAutomatic sequences play important role in computer science and number theory. For a de-tailed account of the theory and applications we refer the reader to the classical monograph [2].One of the applications of such sequences in number theory stems from a celebrated theoremof Christol [5] (also cf. [6]), which asserts that in order to show the transcendence of the powerseries∑n≥1 f(n)zn it is enough to establish that the functionf : N → C is not automatic. Inthis note, rather than working within the general set up, we confine ourselves to functions tak-ing their values in C. There are several equivalent definitions of automatic (or more precisely,q-automatic) sequences. It will be convenient for us to use the following one.Definition 1.1. The sequence f : N → C is called q-automatic if the its q-kernel defined as aset of subsequencesKq(f) ={{f(qin+ r}n≥0| i ≥ 1, 0 ≤ r ≤ qi − 1}is finite.We remark that any q−automatic sequence takes only finitely many values, since it is a func-tion on the states of a finite automation. A function f : N→ C is called completely multiplica-tive if f(mn) = f(m)f(n) for all m,n ∈ N. The question of which multiplicative functions areq-automatic attracted considerable attention of several authors including [15], [14], [3], [13],[12] and [1]. In particular, the following conjecture was made in [3].DEPARTMENT OF MATHEMATICS, KTH ROYAL INSTITUTE OF TECHNOLOGY, STOCKHOLME-mail addresses: lklurman@gmail.com, kurlberg@math.kth.se.Date: December 12, 2019.P.K. was partially supported by the Swedish Research Council (2016-03701).12 AUTOMATIC SEQUENCESConjecture 1.2 (Bell-Bruin-Coons). For any multiplicative q-automatic function f : N → Cthere exists an eventually periodic function g : N→ C, such that f(p) = g(p) for all primes p.This conjecture is still open in general, although some progress has been made when f isassumed to be completely multiplicative. In particular, Schlage-Puchta [14] showed that a com-pletely multiplicative q-automatic sequence which does not vanish is almost periodic. Hu [9]improved on that result by showing that the same conclusion holds under a slightly weakerhypothesis. Our first result confirms a strong form of Conjecture 1.2 when f is additionallyassumed to be completely multiplicative function.Theorem 1.3. Let q ≥ 2 and let f : N → C be a completely multiplicative q-automaticsequence. Then there exists a Dirichlet character χ of conductor Q such that either f(n) =χ(n), for all n ∈ N such that (n,Q) = 1, or f(p) = 0 for all sufficiently large p.Theorem 1.3 also confirms the first part of Conjecture 4.2 in [1]. We remark that a similarresult has been very recently obtained independently by Li [12] using combinatorial methodsrelying on the techniques developed in the theory of automatic sequences. Our proof is shorterand builds upon two deep number theoretic results. Further, assuming the generalized Riemannhypothesis (which in particular implies a strong form of the Artin primitive root conjecturefor primes in progressions) our method can be adapted to show the full conjecture (i.e., theassumption on complete multiplicativity can be removed).Note added in July 2019. While the present article was under submission the authors [10],and independently J. Konieczny [11], found rather different ways to settle the Conjecture 1.2unconditionally. Both approaches use a modification of our Proposition 2.1 to deal with thecase where f vanishes on finitely many primes, but then deviate significantly from the approachtaken here. In [11] the author proceeds by using the structural theory of automatic sequences, inparticular the notion of “arid sets”, to deal with the remaining cases. In our work [10], insteadof using the connection to the Artin conjecture (as in the present paper), we establish certainproperties of the divisors of the sequences {aqn + b}n≥1, for a, b, q ∈ N, to finish up the proof.Acknowledgment. The first author learnt about the question considered here while attendingan excellent talk of J.-P. Allouche at the Institute of Hentri Poincaré (IHP). We are also grate-ful to the referee for valuable comments and suggestions. O.K. would like to thank IHP forexcellent working conditions during his visit.2. PROOF OF THE MAIN RESULTWe begin with a simple albeit important remark. Since f is q-automatic and completelymultiplicative the image of f : N→ C is finite and therefore for any prime p, f(p) = 0 or f(p)is a root of unity.Proposition 2.1. Let f : N → C be a q-automatic completely multiplicative function and letM0 = {p| f(p) = 0} . If |M0| < ∞, then there exists a Dirichlet character χ : N → C suchthat f(p) = χ(p) for all p /∈M0.Proof. Since f is q-automatic there exist positive integers i1 6= i2, such that f(qi1n + 1) =f(qi2n+ 1) for all n ≥ 1. If n = m∏p∈M0 p, thenf(qi1m∏p∈M0 p+ 1)f(qi2m∏p∈M0 p+ 1)= 1 6= 0,AUTOMATIC SEQUENCES 3for all m ≥ 1. The conclusion now immediately follows from Theorem 2 of [7]. Suppose that |M0| = ∞. We are going to show that in this case f(p) = 0 for all sufficientlylarge primes p. Replacing f by |f |, which is also q-automatic, it is enough to prove the claimfor the binary valued f : N→ {0, 1}.Let 1, n = [1, n] ∩ Z. Since f is q-automatic, the q−kernel of f is finite and therefore thereexists k0 = k0(f), such that for all i ≥ 1 and 0 ≤ r ≤ qi − 1, the equalities f(qin+ r) = 0 forn ∈ 1, k0 imply f(qin+ r) = 0 for all n ≥ 1.Lemma 2.2. Suppose that |M0| =∞. For k0 as above and for any p1, p2, . . . , pk0 ∈M0, suchthat pj > max{q, k0} for all j ∈ 1, k0, there exists r = r(q, p1, . . . , pk0) such that (r, qpi) = 1for all i ∈ 1, k0 and f(n∏i≤k0 pi + r) = 0 for all n ≥ 1. We may further assume that r ≡ 3(mod 16), and (r − 1,∏i≤k0 pi) = 1.Proof. For an integer parameter A ≥ logq pk0 , which we shall choose later, by the Chineseremainder theorem there exists rA such that (rA, q) = 1 and rA ≡ −sq2A (mod ps) for alls ∈ 1, k0. Since ps|q2As + rA we have f(q2An + rA) = 0 for all n ∈ 1, k0. The latter impliesthat f(q2An + rA) = 0 for all n ≥ 1. We claim that f(rA) = 0. Indeed, if this is not the casewe choose a prime p, such that f(p) = 1 and consider m = pϕ(q2A)rA, where ϕ(n) denotesthe usual Euler’s totient function. Clearly m ≡ rA (mod q2A) and consequently 0 = f(m) =(f(p))ϕ(q2A)f(rA) = 1, a contradiction. Note, that the same argument works for n∏i≤k0 pi+rAin place of rA and therefore we conclude that f(n∏i≤k0 pi + rA) = 0 for all n ≥ 1. Settingr = rA finishes the proof. Next, without loss of generality we may assume that there exist three sufficiently large primest, t′, t′′ > max(q, k0) such that f(t) = f(t′) = f(t′′) = 1, since otherwise we are done. Wewrite∏a<p≤b p to denote the product of all primes a < p ≤ b. We will require the followingconsequence of a result due to Heath-Brown [8].Lemma 2.3. Given distinct primes t, t′, t′′ > max(q, k0) and r = r(q, p1, . . . , pk0) as inLemma 2.2, there exists infinitely many primes qi ≡ r (mod 16∏i≤k0 pi) such that at leastone of t, t′, t′′ (say t) is a primitive root modulo qi. Moreover, by passing to a subsequence wemay assume that for such primes (qi − 1, qj − 1) = 2 for i 6= j, and for each l ∈ 1, k0 we have(l/qi) = 1 for all i ∈ 1, k0.Proof. Let v = 16(∏i≤k0 pi)∏2<p≤k0 p and chose u such that u ≡ 3 (mod 16) and u ≡ r(mod∏i≤k0 pi), with r as in Lemma 2.2. Moreover, by quadratic reciprocity we may furtherselect u (mod∏2<p≤k0 p) such that (u/p) = 1 for all primes p ≤ k0, and (u−1,∏2<p≤k0 p) =1. In particular, we have (−3/p) = −1 for any prime p ≡ u mod v. Applying Lemma 3 of[8], with u, v as above and k = 1 (and K = 2k = 2) there exists α ∈ (1/4, 1/2] and δ > 0 suchthat|{p ≤ x : p ≡ u (mod v), (p− 1)/K = P2(α, δ)|  x/(log x)2,with the implied constant possibly depending on α, with P2(α, δ) denoting the union of theset of primes, together with the set of almost primes n = t1t2 with t1 < t2 both primes,and t1 ∈ [nα, n1/2−δ]. Heath-Brown’s argument then shows that at least one of t, t′, t′′ is aprimitive root for infinitely many primes p ≡ u (mod v). Whether the primes qi produced havethe properties that (qi − 1)/2 is prime, or that (qi − 1)/2 = t1t2, we may pass to an infinite4 AUTOMATIC SEQUENCESsubsequence of primes q1 < q2 < . . . (satisfying q1 > q) so that (qi − 1, qj − 1) = 2 for i 6= j(for the latter case of almost primes, note that both t1 and t2 are growing.) Proposition 2.4. Suppose that |M0| =∞. Then f(p) = 0 for all sufficiently large primes p.Proof. By Lemma 2.3, we may select prime t with f(t) = 1, which is a primitive root moduloinfinitely primes q1 < q2 · · · < qk0 (satisfying q1 > max(k0, q)) such that qi ≡ r (mod 16∏j≤k0 pk0)and consequently f(qi) = 0. From the proof of Lemma 2.2 it follows that there exists rA, suchthat f(n∏i≤k0 qi + rA) = for all n ≥ 1. Since t is a primitive root modulo qj for j ∈ 1, k0,there exists γj such that tγj ≡ rA (mod qj) for j ∈ 1, k0. By the construction and Lemma 2.3we have (rA/qi) = (−iq2A/qi) = −1 and thus all γi have the same parity. Consequently, bythe Chinese remainder theorem we can choose γ ∈ N, such that γ ≡ γj (mod qj − 1) for allj ∈ 1, k0. For γ defined this way we have tγ ≡ rA (mod∏j≤k0 qj). Hence, f(tγ) must bezero. On the other hand f(tγ) = f(t)γ = 1, and this contradiction finishes the proof. Combining Proposition 2.1 and Proposition 2.4 yields the conclusion of Theorem 1.3REFERENCES[1] J.-P. Allouche and L. Goldmakher. Mock characters and the Kronecker symbol. J. Number Theory, 192:356–372, 2018.[2] J.-P. Allouche and J. Shallit. Automatic sequences. Cambridge University Press, Cambridge, 2003. Theory,applications, generalizations.[3] J. P. Bell, N. Bruin, and M. Coons. Transcendence of generating functions whose coefficients are multiplica-tive. Trans. Amer. Math. Soc., 364(2):933–959, 2012.[4] J. P. Bell, M. Coons, and K. G. Hare. The minimal growth of a k-regular sequence. Bull. Aust. Math. Soc.,90(2):195–203, 2014.[5] G. Christol. Ensembles presque periodiques k-reconnaissables. Theoret. Comput. Sci., 9(1):141–145, 1979.[6] G. Christol, T. Kamae, M. Mendès France, and G. Rauzy. Suites algébriques, automates et substitutions. Bull.Soc. Math. France, 108(4):401–419, 1980.[7] P. D. T. A. Elliott and J. Kish. Harmonic analysis on the positive rationals. Determination of the groupgenerated by the ratios (an+ b)/(An+B). Mathematika, 63(3):919–943, 2017.[8] D. R. Heath-Brown. Artin’s conjecture for primitive roots. Quart. J. Math. Oxford Ser. (2), 37(145):27–38,1986.[9] Y. Hu. Subword complexity and non-automaticity of certain completely multiplicative functions. Adv. in Appl.Math., 84:73–81, 2017.[10] O. Klurman and P. Kurlberg. A note on multiplicative automatic sequences, II. arXiv:1905.10897, 2019.[11] J. Konieczny. On multiplicative automatic sequences. arXiv:1905.11981, 2019.[12] S. Li. On completely multiplicative automatic sequences. arXiv:1903.04385, 2019.[13] J.-C. Schlage-Puchta. A criterion for non-automaticity of sequences. J. Integer Seq., 6(3):Article 03.3.8, 5,2003.[14] J.-C. Schlage-Puchta. Completely multiplicative automatic functions. Integers, 11:A31, 8, 2011.[15] S. Yazdani. Multiplicative functions and k-automatic sequences. J. Théor. Nombres Bordeaux, 13(2):651–658, 2001.

We prove that any 푞 ‐automatic multiplicative function 푓:ℕ→ℂ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of Bell, Bruin and Coons [Trans. Amer. Math. Soc. 364 (2012) 933–959]

                          Klurman, O., & Kurlberg, P. (2020). A note on multiplicative automaticsequences, II. Bull. Lond. Math. Soc., 52(1), 185-188.https://doi.org/10.1112/blms.12318Peer reviewed versionLink to published version (if available):10.1112/blms.12318Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia Wiley at https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12318 . Please refer to anyapplicable terms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/user-guides/explore-bristol-research/ebr-terms/A NOTE ON MULTIPLICATIVE AUTOMATIC SEQUENCES, IIOLEKSIY KLURMAN AND PA¨R KURLBERGABSTRACT. We prove that any q-automatic multiplicative function f :N → C either essentially coincides with a Dirichlet character, or van-ishes on all sufficiently large primes. This confirms a strong form of aconjecture of J. Bell, N. Bruin, and M. Coons.1. INTRODUCTIONAutomatic sequences play important role in computer science and num-ber theory. For a detailed account of the theory and applications we referthe reader to the classical monograph [2]. One of the applications of suchsequences in number theory stems from a celebrated theorem of Christol [5](also cf. [6]), which asserts that in order to show the transcendence of thepower series∑n≥1 f(n)zn over Fq[z] it is enough to establish that the func-tion f : N → Fq is not q-automatic. In this note, rather than workingwithin the general set up, we confine ourselves to functions with range inC. There are several equivalent definitions of automatic (or more precisely,q-automatic) sequences. It will be convenient for us to use the followingone.Definition 1.1. The sequence f : N → C is called q-automatic if the q-kernel of it, defined as a set of subsequencesKq(f) ={{f(qin+ r)}n≥0| i ≥ 1, 0 ≤ r ≤ qi − 1} ,is finite.We remark that any q−automatic sequence takes only finitely many val-ues, since it is a function on the states of finite automata. A functionf : N → C is called multiplicative if f(mn) = f(m)f(n) for all pairs(m,n) = 1. The question of which multiplicative functions are q-automaticattracted considerable attention of several authors including [17], [15], [7],[16], [3], [1], [14], [12]. In particular, the following conjecture was madein [3].Conjecture 1.2 (Bell-Bruin-Coons). For any multiplicative q-automatic func-tion f : N → C there exists an eventually periodic function g : N → C,such that f(p) = g(p) for all primes p.DEPARTMENT OF MATHEMATICS, KTH ROYAL INSTITUTE OF TECHNOLOGY,STOCKHOLME-mail addresses: lklurman@gmail.com, kurlberg@math.kth.se.Date: Wednesday 11th December, 2019.P.K. was partially supported by the Swedish Research Council (2016-03701).12 AUTOMATIC SEQUENCESSome progress towards this conjecture has been made when f is assumedto be completely multiplicative. In particular, Schlage-Puchta [16] showedthat a completely multiplicative q-automatic sequence which does not van-ish is almost periodic. Hu [10] improved on that result by showing that thesame conclusion holds under a slightly weaker hypothesis. Allouche andGoldmakher [1] considered a related question classifying what they called“mock” Dirichlet characters. Finally Li [14] and the authors [12] very re-cently proved Conjecture 1.2 when f is additionally assumed to be com-pletely multiplicative. Moreover, the methods of [12] settled Conjecture 1.2fully under the assumption of the Generalized Riemann Hypothesis. Thekey tool there was a variant of Heath-Brown’s results [9] on Artin’s primi-tive root conjecture. In this article, we develop an alternative approach, andunconditionally prove a strong form of Conjecture 1.2.Theorem 1.3. Let q ≥ 2 and let f : N → C be multiplicative q-automaticsequence. Then, there exists a Dirichlet character χ and an integer Q ≥ 1,such that either f(n) = χ(n), for all (n,Q) = 1 or f(p) = 0 for allsufficiently large p.We remark that J. Konieczny [13], very recently and independently es-tablished a variant of Theorem 1.3 using different methods relying on thestructure theory of automatic sequences.Acknowledgments. The authors would like to thank J.-P. Allouche fromwhom they learnt about the question considered in this note and for hisinterest and encouragement. We are also very grateful to the referee forpointing out the possibility that our original argument can be simplified, aswell as for valuable comments and suggestions.2. PROOF OF THE MAIN RESULTLet P0 the set of primes for which f(p) = 0. Since f is q-automatic, it iswell-known that the image of f : N→ C is finite and therefore if we definethe setsP>1 = {p prime : |f(pe)| > 1 for some e ≥ 1}andP<1 = {p prime : |f(pe)| < 1 for some e ≥ 1},then, from multiplicativity of f we easily deduce |P>1|, |P<1| < ∞. Webegin by assuming additionally that |P0| <∞.Proposition 2.1. Let f : N → C be a q-automatic multiplicative functionand suppose that |P0| <∞. Then there exists a Dirichlet character χ, suchthat f(pk) = χ(pk) for all p /∈ P>1 ∪ P<1 ∪ P0 and any k ≥ 1.Proof. Since f is q-automatic, we have that the kernel Kq(f) is finite. Bythe pigeonhole principle, there exist positive integers i1 6= i2 such thatAUTOMATIC SEQUENCES 3f(qi1n+ 1) = f(qi2n+ 1) for all n ≥ 1. If n = m∏p∈P>1∪P<1∪P0 p then(2.1)f(qi1m∏p∈P>1∪P<1∪P0 p+ 1)f(qi2m∏p∈P>1∪P<1∪P0 p+ 1)= 1 6= 0,for all m ≥ 1. The conclusion now immediately follows from the followingresult by Elliott and Kish (stated as Theorem 1 in the arXiv preprint.)Theorem 2.2 ([8], Theorem 2). Let integers a > 0, b, B, satisfy ∆ = aB −Ab 6= 0. Set δ = 6(a,A)(aA)2∆3. If a multiplicative complex-valuedfunction g satisfiesg(an+ bAn+B)= c 6= 0on all but finitely positive integers, n, then there is a Dirichlet character(mod δ) with which g coincides on all primes that do not divide δ.We remark that the result is stated only for completely multiplicativefunctions, but the proof is in fact also valid for multiplicative functions.Alternatively, one could also apply correlation formulas developed in [11]to get the result. In what follows we will assume that |P0| =∞ and show that in this casef(p) = 0 for all sufficiently large primes p. We begin with a convenientreduction: defining g asg(n) ={1, if f(n) 6= 0,0, if f(n) = 0,we note that sequence (g(n))n is also q-automatic, and has the same zeroset as the sequence (f(n))n. Thus, replacing f by g, it is therefore enoughto prove the claim for binary valued functions f : N→ {0, 1}.To facilitate our discussion, we introduce the setP̂1 := {p prime : f(pe) = 1 for some e ≥ 1}.Since the case of finite P̂1 is trivial, we from now on assume that P̂1 isinfinite.We note that if f is q-automatic, then there exists k0 = k0(f) with thefollowing property: if for some i ≥ 1 and 0 ≤ r ≤ qi − 1 the equalityf(qin+ r) = 0 holds for all integer n ∈ [1, k0], then f(qin+ r) = 0 for alln ≥ 1. A useful consequence is the following:Corollary 2.3. Given any arithmetic progression {qlm + r}m≥1, with r <ql, so that f(qlm+ r) = 1 for infinitely many m, there exists m0 ≤ k0 suchthat f(qlm0 + r) = 1.We next show that non-vanishing of f in certain progressions is eithertrivial or infinite.Lemma 2.4. Assume that |P̂1| =∞. Given r, l ∈ N such that f(r) = 1 andql > r, there exists infinitely many m such that f(qlm+ r) = 1.4 AUTOMATIC SEQUENCESProof. Let pi ∈ P̂1 denote distinct primes such that pi > r and select ex-ponents ei so that f(peii ) = 1, and denote Pi = peii . We may then selecta sequence of finite disjoint sets Ij such that∏i∈Ij Pi ≡ 1 mod ql. Let-ting nj = r ·∏i∈Ij Pi we find, using multiplicativity and that (r, Pi) = 1,that f(nj) = 1, and that nj ≡ r mod ql. Consequently we can writenj = qlmj + r, and the proof is concluded. Let q1, . . . , qk0+1 ∈ P0 denote distinct primes with qi > q for all 1 ≤i ≤ k0 + 1. Define Q :=∏k0+1i=1 qi. By the Chinese remainder theorem,there exists an interval I ⊂ Z/Q2Z with |I| = k0 + 1 such that all integersn ∈ I have the property that at least one qi||n. In particular, f(n) = 0 forall n ∈ I . Further, letting L = φ(Q2), we note that qL ≡ 1 mod Q2.We can now arrive at a contradiction by constructing a sequence of inte-ger ri’s so that on one hand f(ri) = 1 for all i, while at the same time, someri mod Q2 must fall in I and consequently f(ri) = 0.To construct such a sequence we argue as follows. Given r1 such thatf(r1) = 1, select s1 so that l1 := s1L has the property that ql1 > r. ByLemma 2.4, there exist infinitely many m such that f(ql1m + r1) = 1, andthus, by Corollary 2.3, there exists m1 ≤ k0 such thatr2 := ql1m1 + r1has the property that f(r2) = 1. We also note that r2 ≡ m1 + r1 mod Q2.Proceeding inductively, given ri with properties as above, we can defineri+1 := qlimi + riso that f(ri+1) = 1, mi ≤ k0, and ri+1 ≡ ri + mi mod Q2 (in particular,take li = Lsi with si large enough so that qli > ri.)Now, as all mi ≤ k0, we find that {ri mod Q2} must intersect I , conse-quently f(ri) = 0 for some i and we have arrived at a contradiction.Remark 2.5. The proof can be modified to give a slightly stronger conclu-sion, namely if |P0| = ∞ then |P̂1| < ∞. From here one can easily getmore refined information about the set of values of {f(pn)}n≥1. We leavethe details to the interested reader.REFERENCES[1] J.-P. Allouche and L. Goldmakher. Mock characters and the Kronecker symbol. J.Number Theory, 192:356–372, 2018.[2] J.-P. Allouche and J. Shallit. Automatic sequences. Cambridge University Press, Cam-bridge, 2003. Theory, applications, generalizations.[3] J. P. Bell, N. Bruin, and M. Coons. Transcendence of generating functions whosecoefficients are multiplicative. Trans. Amer. Math. Soc., 364(2):933–959, 2012.[4] J. P. Bell, M. Coons, and K. G. Hare. The minimal growth of a k-regular sequence.Bull. Aust. Math. Soc., 90(2):195–203, 2014.[5] G. Christol. Ensembles presque periodiques k-reconnaissables. Theoret. Comput.Sci., 9(1):141–145, 1979.AUTOMATIC SEQUENCES 5[6] G. Christol, T. Kamae, M. Mende`s France, and G. Rauzy. Suites alge´briques, auto-mates et substitutions. Bull. Soc. Math. France, 108(4):401–419, 1980.[7] M. Coons. (Non)automaticity of number theoretic functions. J. The´or. Nombres Bor-deaux, 22(2):339–352, 2010.[8] P. D. T. A. Elliott and J. Kish. Harmonic analysis on the positive rationals. Deter-mination of the group generated by the ratios (an + b)/(An + B). Mathematika,63(3):919–943, 2017.[9] D. R. Heath-Brown. Artin’s conjecture for primitive roots. Quart. J. Math. OxfordSer. (2), 37(145):27–38, 1986.[10] Y. Hu. Subword complexity and non-automaticity of certain completely multiplica-tive functions. Adv. in Appl. Math., 84:73–81, 2017.[11] O. Klurman. Correlations of multiplicative functions and applications. Compos.Math., 153(8):1622–1657, 2017.[12] O. Klurman and P. Kurlberg. A note on multiplicative automatic sequences. C. R.Math. Acad. Sci. Paris, 357(10):752–755, 2019.[13] J. Konieczny. On multiplicative automatic sequences. arXiv:1905.11981, 2019.[14] S. Li. On completely multiplicative automatic sequences. arXiv:1903.04385, 2019.[15] J.-C. Schlage-Puchta. A criterion for non-automaticity of sequences. J. Integer Seq.,6(3):Article 03.3.8, 5, 2003.[16] J.-C. Schlage-Puchta. Completely multiplicative automatic functions. Integers,11:A31, 8, 2011.[17] S. Yazdani. Multiplicative functions and k-automatic sequences. J. The´or. NombresBordeaux, 13(2):651–658, 2001.

A note on multiplicative automatic sequences, II

We prove that any $q$-automatic multiplicative function $f:\mathbb{N}\to\mathbb{C}$ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of J. Bell, N. Bruin, and M. Coons

Klurman, O.

Kurlberg, P.

MPG.PuRe

Bull. London Math. Soc. 52 (2020) 185–188 doi:10.1112/blms.12318A note on multiplicative automatic sequences, IIOleksiy Klurman and Pär KurlbergAbstractWe prove that any q-automatic multiplicative function f : N → C either essentially coincideswith a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strongform of a conjecture of Bell, Bruin and Coons [Trans. Amer. Math. Soc. 364 (2012) 933–959].1. IntroductionAutomatic sequences play important role in computer science and number theory. For a detailedaccount of the theory and applications, we refer the reader to the classical monograph [2]. Oneof the applications of such sequences in number theory stems from a celebrated theorem ofChristol [5] (also cf. [6]), which asserts that in order to show the transcendence of the powerseries∑n1 f(n)zn over Fq[z] it is enough to establish that the function f : N → Fq is notq-automatic. In this note, rather than working within the general set up, we confine ourselvesto functions with range in C. There are several equivalent definitions of automatic (or moreprecisely, q-automatic) sequences. It will be convenient for us to use the following one.Definition 1.1. The sequence f : N → C is called q-automatic if the q-kernel of it, definedas a set of subsequencesKq(f) ={{f(qin + r)}n0| i  1, 0  r  qi − 1},is finite.We remark that any q-automatic sequence takes only finitely many values, since it is afunction on the states of finite automata. A function f : N → C is called multiplicative iff(mn) = f(m)f(n) for all pairs (m,n) = 1. The question of which multiplicative functions areq-automatic attracted considerable attention of several authors including [1, 3, 7, 12, 14–17].In particular, the following conjecture was made in [3].Conjecture 1.2 (Bell–Bruin–Coons). For any multiplicative q-automatic functionf : N→C, there exists an eventually periodic function g : N → C, such that f(p) = g(p) for allprimes p.Some progress towards this conjecture has been made when f is assumed to be completelymultiplicative. In particular, Schlage-Puchta [16] showed that a completely multiplicative q-automatic sequence which does not vanish is almost periodic. Hu [10] improved on that resultby showing that the same conclusion holds under a slightly weaker hypothesis. Allouche andGoldmakher [1] considered a related question classifying what they called ‘mock’ DirichletReceived 6 June 2019; revised 26 November 2019; published online 23 January 2020.2010 Mathematics Subject Classification 11B85 (primary), 68R15 (secondary).P. K. was partially supported by the Swedish Research Council (2016-03701).Correction added on November 6, 2020, after first online publication: Max-Planck-Gesellschaft zur Förderungder Wissenschaften e.V. was added for Oleksiy Klurman.C 2020 The Author(s). The Bulletin of the London Mathematical Society is copyright C LondonMathematical Society. This is an open access article under the terms of the Creative Commons AttributionLicense, which permits use, distribution and reproduction in any medium, provided the original work is properlycited.186 OLEKSIY KLURMAN AND PÄR KURLBERGcharacters. Finally Li [14] and the authors [12] very recently proved Conjecture 1.2 when fis additionally assumed to be completely multiplicative. Moreover, the methods of [12] settledConjecture 1.2 fully under the assumption of the Generalized Riemann Hypothesis. The keytool there was a variant of Heath-Brown’s results [9] on Artin’s primitive root conjecture. Inthis article, we develop an alternative approach, and unconditionally prove a strong form ofConjecture 1.2.Theorem 1.3. Let q  2 and let f : N → C be multiplicative q-automatic sequence. Then,there exists a Dirichlet character χ and an integer Q  1, such that either f(n) = χ(n), for all(n,Q) = 1 or f(p) = 0 for all sufficiently large p.We remark that Konieczny [13], very recently and independently established a variant ofTheorem 1.3 using different methods relying on the structure theory of automatic sequences.2. Proof of the main resultLet P0 the set of primes for which f(p) = 0. Since f is q-automatic, it is well known that theimage of f : N → C is finite and therefore if we define the setsP>1 = {p prime : |f(pe)| > 1 for some e  1}andP<1 = {p prime : |f(pe)| < 1 for some e  1},then, from multiplicativity of f , we easily deduce |P>1|, |P<1| < ∞. We begin by assumingadditionally that |P0| < ∞.Proposition 2.1. Let f : N → C be a q-automatic multiplicative function and suppose|P0| < ∞. Then there exists a Dirichlet character χ, such that f(pk) = χ(pk) for all p /∈ P>1 ∪P<1 ∪ P0 and any k  1.Proof. Since f is q-automatic, we have that the kernel Kq(f) is finite. By the pigeonholeprinciple, there exist positive integers i1 = i2 such that f(qi1n + 1) = f(qi2n + 1) for all n  1.If n = m∏p∈P>1∪P<1∪P0 p, thenf⎛⎝qi1m ∏p∈P>1∪P<1∪P0p + 1⎞⎠f⎛⎝qi2m ∏p∈P>1∪P<1∪P0p + 1⎞⎠ = 1 = 0, (2.1)for all m  1. The conclusion now immediately follows from the following result by Elliott andKish (stated as Theorem 1 in the arXiv preprint.)Theorem 2.2 [8, Theorem 2]. Let integers a > 0, b, B, satisfy Δ = aB −Ab = 0. Setδ = 6(a,A)(aA)2Δ3. If a multiplicative complex-valued function g satisfiesg(an + bAn + B)= c = 0on all but finitely positive integers, n, then there is a Dirichlet character (mod δ) with whichg coincides on all primes that do not divide δ.AUTOMATIC SEQUENCES 187We remark that the result is stated only for completely multiplicative functions, but the proofis in fact also valid for multiplicative functions. Alternatively, one could also apply correlationformulas developed in [11] to get the result. In what follows we will assume that |P0| = ∞ and show that in this case f(p) = 0 for allsufficiently large primes p. We begin with a convenient reduction: defining g asg(n) ={1, if f(n) = 0,0, if f(n) = 0,we note that sequence (g(n))n is also q-automatic, and has the same zero set as the sequence(f(n))n. Thus, replacing f by g, it is therefore enough to prove the claim for binary valuedfunctions f : N → {0, 1}.To facilitate our discussion, we introduce the setP̂1 := {p prime : f(pe) = 1 for some e  1}.Since the case of finite P̂1 is trivial, we from now on assume that P̂1 is infinite.We note that if f is q-automatic, then there exists k0 = k0(f) with the following property: iffor some i  1 and 0  r  qi − 1 the equality f(qin + r) = 0 holds for all integer n ∈ [1, k0],then f(qin + r) = 0 for all n  1. A useful consequence is the following:Corollary 2.3. Given any arithmetic progression {qlm + r}m1, with r < ql, so thatf(qlm + r) = 1 for infinitely many m, there exists m0  k0 such that f(qlm0 + r) = 1.We next show that non-vanishing of f in certain progressions is either trivial or infinite.Lemma 2.4. Assume |P̂1| = ∞. Given r, l ∈ N such that f(r) = 1 and ql > r, there existsinfinitely many m such that f(qlm + r) = 1.Proof. Let pi ∈ P̂1 denote distinct primes such that pi > r and select exponents ei so thatf(peii ) = 1, and denote Pi = peii . We may then select a sequence of finite disjoint sets Ij suchthat∏i∈Ij Pi ≡ 1 mod ql. Letting nj = r ·∏i∈Ij Pi we find, using multiplicativity and that(r, Pi) = 1, that f(nj) = 1, and that nj ≡ r mod ql. Consequently, we can write nj = qlmj + r,and the proof is concluded. Let q1, . . . , qk0+1 ∈ P0 denote distinct primes with qi > q for all 1  i  k0 + 1. DefineQ :=∏k0+1i=1 qi. By the Chinese remainder theorem, there exists an interval I ⊂ Z/Q2Z with|I| = k0 + 1 such that all integers n ∈ I have the property that at least one qi||n. In particular,f(n) = 0 for all n ∈ I. Further, letting L = φ(Q2), we note that qL ≡ 1 mod Q2.We can now arrive at a contradiction by constructing a sequence of integer functions ri sothat on one hand f(ri) = 1 for all i, while at the same time, some ri mod Q2 must fall in Iand consequently f(ri) = 0.To construct such a sequence we argue as follows. Given r1 such that f(r1) = 1, select s1so that l1 := s1L has the property that ql1 > r. By Lemma 2.4, there exist infinitely many msuch that f(ql1m + r1) = 1, and thus, by Corollary 2.3, there exists m1  k0 such thatr2 := ql1m1 + r1has the property that f(r2) = 1. We also note that r2 ≡ m1 + r1 mod Q2.Proceeding inductively, given ri with properties as above, we can defineri+1 := qlimi + riso that f(ri+1) = 1, mi  k0, and ri+1 ≡ ri + mi mod Q2 (in particular, take li = Lsi with silarge enough so that qli > ri.)Now, as all mi  k0, we find that {ri mod Q2} must intersect I, consequently f(ri) = 0 forsome i and we have arrived at a contradiction.188 OLEKSIY KLURMAN AND PÄR KURLBERGRemark 2.5. The proof can be modified to give a slightly stronger conclusion, namely if|P0| = ∞, then |P̂1| < ∞. From here one can easily get more refined information about the setof values of {f(pn)}n1. We leave the details to the interested reader.Acknowledgements. The authors would like to thank J.-P. Allouche from whom they learntabout the question considered in this note and for his interest and encouragement. We are alsovery grateful to the referee for pointing out the possibility that our original argument can besimplified, as well as for valuable comments and suggestions.Open access funding enabled and organized by Projekt DEAL.References1. J.-P. Allouche and L. Goldmakher, ‘Mock characters and the Kronecker symbol’, J. Number Theory192 (2018) 356–372.2. J.-P. Allouche and J. Shallit, Automatic sequences: theory, applications, generalizations (CambridgeUniversity Press, Cambridge, 2003).3. J. P. Bell, N. Bruin and M. Coons, ‘Transcendence of generating functions whose coefficients aremultiplicative’, Trans. Amer. Math. Soc. 364 (2012) 933–959.4. J. P. Bell, M. Coons and K. G. Hare, ‘The minimal growth of a k-regular sequence’, Bull. Aust. Math.Soc. 90 (2014) 195–203.5. G. Christol, ‘Ensembles presque periodiques k-reconnaissables’, Theoret. Comput. Sci. 9 (1979) 141–145.6. G. Christol, T. Kamae, M. Mendès France and G. Rauzy, ‘Suites algébriques, automates etsubstitutions’, Bull. Soc. Math. France 108 (1980) 401–419.7. M. Coons, ‘(Non)automaticity of number theoretic functions’, J. Théor. Nombres Bordeaux 22 (2010)339–352.8. P. D. T. A. Elliott and J. Kish, ‘Harmonic analysis on the positive rationals. Determination of thegroup generated by the ratios (an + b)/(An + B)’, Mathematika 63 (2017) 919–943.9. D. R. Heath-Brown, ‘Artin’s conjecture for primitive roots’, Q. J. Math. 37 (1986) 27–38.10. Y. Hu, ‘Subword complexity and non-automaticity of certain completely multiplicative functions’, Adv.Appl. Math. 84 (2017) 73–81.11. O. Klurman, ‘Correlations of multiplicative functions and applications’, Compos. Math. 153 (2017) 1622–1657.12. O. Klurman and P. Kurlberg, ‘A note on multiplicative automatic sequences’, C. R. Math. Acad. Sci.Paris 357 (2019) 752–755.13. J. Konieczny, ‘On multiplicative automatic sequences’, Bull. London Math. Soc. 52 (2020) 175–184.14. S. Li, ‘On completely multiplicative automatic sequences’, Preprint, 2019, arXiv:1903.04385.15. J.-C. Schlage-Puchta, ‘A criterion for non-automaticity of sequences’, J. Integer Seq. 6 (2003) 03.3.8.16. J.-C. Schlage-Puchta, ‘Completely multiplicative automatic functions’, Integers 11 (2011) 731–738.17. S. Yazdani, ‘Multiplicative functions and k-automatic sequences’, J. Théor. Nombres Bordeaux 13 (2001)651–658.Oleksiy KlurmanDepartment of MathematicsKTH Royal Institute of TechnologyStockholm SE-10044SwedenandMax-Planck-Gesellschaft zur Förderung derWissenschaften e.V.Bonn 53111Germanylklurman@gmail.comPär KurlbergDepartment of MathematicsKTH Royal Institute of TechnologyStockholm SE-10044Swedenkurlberg@math.kth.seThe Bulletin of the London Mathematical Society is wholly owned and managed by the London MathematicalSociety, a not-for-profit Charity registered with the UK Charity Commission. All surplus income from itspublishing programme is used to support mathematicians and mathematics research in the form of researchgrants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.

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A note on multiplicative automatic sequences

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