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HARMONIC ANALYSIS ON THE POSITIVE RATIONALS. DETERMINATION OF THE GROUP GENERATED BY THE RATIOS $(an+b)/(An+B)$

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Abelian groups Exponential sums and character sums Multiplicative number theory

Published online by Cambridge University Press:  29 November 2017

P. D. T. A. Elliott  and
Jonathan Kish
P. D. T. A. Elliott
Affiliation:
Department of Mathematics, University of Colorado Boulder, Boulder, Colorado 80309-0395, U.S.A. email pdtae@euclid.colorado.edu
Jonathan Kish
Affiliation:
Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309-0526, U.S.A. email jonathan.kish@colorado.edu
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Abstract

The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture.

MSC classification

Primary: 11K65: Arithmetic functions 11K70: Harmonic analysis and almost periodicity 11K99: None of the above, but in this section 20K99: None of the above, but in this section 11L05: Gauss and Kloosterman sums; generalizations 11L99: None of the above, but in this section 11N64: Other results on the distribution of values or the characterization of arithmetic functions 11N99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 919 - 943
Copyright
Copyright © University College London 2017 

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References

Berrizbeitia, P. and Elliott, P. D. T. A., On products of shifted primes. Ramanujan J. 2(1–2) 1998, 219–223. Paul Erdős memorial volume.CrossRefGoogle Scholar
Britton, J. L., Solution of the word problem for certain types of groups I. Proc. Glasgow Math. Assoc. 3 1956, 45–54.CrossRefGoogle Scholar
Dress, F. and Volkmann, B., Ensembles d’unicité pour les fonctions arithmétiques additives ou multiplicatives. C. R. Acad. Sci. Paris Sér. A-B 287(2) 1978, A43–A46.Google Scholar
Elliott, P. D. T. A., A conjecture of Kátai. Acta Arith. 26(1) 1974/75, 11–20.CrossRefGoogle Scholar
Elliott, P. D. T. A., The law of large numbers for additive arithmetic functions. Math. Proc. Cambridge Philos. Soc. 78(1) 1975, 33–71.CrossRefGoogle Scholar
Elliott, P. D. T. A., General asymptotic distributions for additive arithmetic functions. Math. Proc. Cambridge Philos. Soc. 79(1) 1976, 43–54.CrossRefGoogle Scholar
Elliott, P. D. T. A., Sums and differences of additive arithmetic functions in mean square. J. Reine Angew. Math. 309 1979, 21–54.Google Scholar
Elliott, P. D. T. A., Arithmetic Functions and Integer Products (Grundlehren Math. Wiss. 272 ), Springer (New York, 1985).CrossRefGoogle Scholar
Elliott, P. D. T. A., Multiplicative functions on arithmetic progressions. Mathematika 34(2) 1987, 199–206.CrossRefGoogle Scholar
Elliott, P. D. T. A., A localized Erdős–Wintner theorem. Pacific J. Math. 135(2) 1988, 287–297.CrossRefGoogle Scholar
Elliott, P. D. T. A., Multiplicative functions |g|⩽1 and their convolutions: an overview. In Séminaire de Théorie des Nombres, Paris 1987–88 (Progress in Mathematics 81 ) (ed. Goldstein, C.), Birkhäuser (Boston, 1990), 63–73.Google Scholar
Elliott, P. D. T. A., On the correlation of multiplicative functions. Notas Soc. Mat. Chile 11(1) 1992, 1–11.Google Scholar
Elliott, P. D. T. A., Multiplicative functions on arithmetic progressions. VI. More middle moduli. J. Number Theory 44(2) 1993, 178–208.CrossRefGoogle Scholar
Elliott, P. D. T. A., On the correlation of multiplicative and the sum of additive arithmetic functions. Mem. Amer. Math. Soc. 112(538) 1994, viii+88.Google Scholar
Elliott, P. D. T. A., The multiplicative group of rationals generated by the shifted primes, I. J. Reine Angew. Math. 463 1995, 169–216.Google Scholar
Elliott, P. D. T. A., Duality in Analytic Number Theory (Cambridge Tracts in Mathematics 122 ), Cambridge University Press (Cambridge, 1997).CrossRefGoogle Scholar
Elliott, P. D. T. A., The multiplicative group of rationals generated by the shifted primes, II. J. Reine Angew. Math. 519 2000, 59–71.Google Scholar
Elliott, P. D. T. A., Product representations by rationals. In Number Theoretic Methods: Future Trends, Proceedings of the Second China–Japan Seminar, Iizuka, Japan, March 12–16, 2001 (Dev. Math. 8 ) (eds Kanemitsu, S. and Jia, C.), Kluwer Academic Publishers (Dordrecht, 2002), 119–150.CrossRefGoogle Scholar
Elliott, P. D. T. A., The value distribution of additive arithmetic functions on a line. J. Reine Angew. Math. 642 2010, 57–108.Google Scholar
Kátai, I., On sets characterizing number-theoretical functions. Acta Arith. 13(3) 1968, 315–320.CrossRefGoogle Scholar
Meyer, J., Ensembles d’unicité pour les fonctions additives. Étude analogue dans le cas des fonctions multiplicatives. In Proceedings of the Journées de Théorie Analytique et Elémentaire des Nombres, Université de Paris-Sud, Orsay, France, June 2–3, 1980, Vol. 81 (Publ. Math. Orsay 1 ), Université de Paris-Sud (Orsay, 1981), 19–29.Google Scholar
Stepanauskas, G., The mean values of multiplicative functions, V. In Analytic and Probabilistic Methods in Number Theory, Proceedings of the Third International Conference in Honour of J. Kubilius, Palanga, Lithuania, 24–28 September, 2001 (eds Dubrickas, A., Laurinčikas, A. and Manstavičius, E.), TEV (Vilnius, 2002), 272–281.Google Scholar
Stillwell, J., The word problem and the isomorphism problem for groups. Bull. Amer. Math. Soc. (N.S.) 6(1) 1982, 33–56.Google Scholar
Tao, T., The logarithmic averaged Chowla and Elliott conjectures for two-point correlations. Preprint, 2015, arXiv:1509.05422v2.CrossRefGoogle Scholar
Wolke, D., Bemerkungen über Eindeutigkeitsmengen additiver Funktionen. Elem. Math. 33(1) 1978, 14–16.Google Scholar