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Rigidity theorems for multiplicative functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2018 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 372, no 1-2, p. 651-697Article in journal (Refereed) Published
Abstract [en]

We establish several results concerning the expected general phenomenon that, given a multiplicative function f: N→ C, the values of f(n) and f(n+ a) are “generally” independent unless f is of a “special” form. First, we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values. This implies the solution of the following folklore conjecture: for any completely multiplicative function f: N→ T we have lim infn→∞|f(n+1)-f(n)|=0.Second, we settle an old conjecture due to Chudakov (On the generalized characters. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, p. 487. Gauthier-Villars, Paris) that states that any completely multiplicative function f: N→ C that: (a) takes only finitely many values, (b) vanishes at only finitely many primes, and (c) has bounded discrepancy, is a Dirichlet character. This generalizes previous work of Tao on the Erdős Discrepancy Problem. Finally, we show that if many of the binary correlations of a 1-bounded multiplicative function are asymptotically equal to those of a Dirichlet character χ mod q then f(n) = χ′(n) nit for all n, where χ′ is a Dirichlet character modulo q and t∈ R. This establishes a variant of a conjecture of H. Cohn for multiplicative arithmetic functions. The main ingredients include the work of Tao on logarithmic Elliott conjecture, correlation formulas for pretentious multiplicative functions developed earlier by the first author and Szemeredi’s theorem for long arithmetic progressions. 

Place, publisher, year, edition, pages
Springer New York LLC , 2018. Vol. 372, no 1-2, p. 651-697
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-236652DOI: 10.1007/s00208-018-1724-6ISI: 000445199600021Scopus ID: 2-s2.0-85050003389OAI: oai:DiVA.org:kth-236652DiVA, id: diva2:1262837
Note

Correspondence Address: Mangerel, A.P.; Department of Mathematics, University of TorontoCanada; email: sacha.mangerel@mail.utoronto.ca; Funding details: Fields Institute for Research in Mathematical Sciences; Funding text: Acknowledgements We would like to thank John Friedlander and Andrew Granville for all their advice and encouragement. We are grateful to Andrew Granville and Terence Tao for their valuable comments on an earlier version of the present paper. We thank Mei-Chu Chang for her interest in the results of this paper. We are indebted to Sergey Konyagin and Terence Tao for introducing us to Chudakov’s conjecture. We also thank Imre Kátai for referring us to his conjecture with Subbarao. We take this opportunity to thank the anonymous referee for a careful reading of the paper and for many helpful suggestions leading to an improvement in the exposition of this paper. Finally, we would like to thank both the Mathematical Sciences Research Institute in Berkeley, California, as well as the Fields Institute for Research in the Mathematical Sciences in Toronto for providing excellent working conditions. QC 20181210

Available from: 2018-11-13 Created: 2018-11-13 Last updated: 2018-12-10Bibliographically approved

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