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

  • P. D. T. A. Elliott (a1) and Jonathan Kish (a2)
Abstract

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

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References
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1. Berrizbeitia, P. and Elliott, P. D. T. A., On products of shifted primes. Ramanujan J. 2(1–2) 1998, 219223. Paul Erdős memorial volume.
2. Britton, J. L., Solution of the word problem for certain types of groups I. Proc. Glasgow Math. Assoc. 3 1956, 4554.
3. 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, A43A46.
4. Elliott, P. D. T. A., A conjecture of Kátai. Acta Arith. 26(1) 1974/75, 1120.
5. Elliott, P. D. T. A., The law of large numbers for additive arithmetic functions. Math. Proc. Cambridge Philos. Soc. 78(1) 1975, 3371.
6. Elliott, P. D. T. A., General asymptotic distributions for additive arithmetic functions. Math. Proc. Cambridge Philos. Soc. 79(1) 1976, 4354.
7. Elliott, P. D. T. A., Sums and differences of additive arithmetic functions in mean square. J. Reine Angew. Math. 309 1979, 2154.
8. Elliott, P. D. T. A., Arithmetic Functions and Integer Products (Grundlehren Math. Wiss. 272 ), Springer (New York, 1985).
9. Elliott, P. D. T. A., Multiplicative functions on arithmetic progressions. Mathematika 34(2) 1987, 199206.
10. Elliott, P. D. T. A., A localized Erdős–Wintner theorem. Pacific J. Math. 135(2) 1988, 287297.
11. 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), 6373.
12. Elliott, P. D. T. A., On the correlation of multiplicative functions. Notas Soc. Mat. Chile 11(1) 1992, 111.
13. Elliott, P. D. T. A., Multiplicative functions on arithmetic progressions. VI. More middle moduli. J. Number Theory 44(2) 1993, 178208.
14. 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.
15. Elliott, P. D. T. A., The multiplicative group of rationals generated by the shifted primes, I. J. Reine Angew. Math. 463 1995, 169216.
16. Elliott, P. D. T. A., Duality in Analytic Number Theory (Cambridge Tracts in Mathematics 122 ), Cambridge University Press (Cambridge, 1997).
17. Elliott, P. D. T. A., The multiplicative group of rationals generated by the shifted primes, II. J. Reine Angew. Math. 519 2000, 5971.
18. 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), 119150.
19. Elliott, P. D. T. A., The value distribution of additive arithmetic functions on a line. J. Reine Angew. Math. 642 2010, 57108.
20. Kátai, I., On sets characterizing number-theoretical functions. Acta Arith. 13(3) 1968, 315320.
21. 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), 1929.
22. 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), 272281.
23. Stillwell, J., The word problem and the isomorphism problem for groups. Bull. Amer. Math. Soc. (N.S.) 6(1) 1982, 3356.
24. Tao, T., The logarithmic averaged Chowla and Elliott conjectures for two-point correlations. Preprint, 2015, arXiv:1509.05422v2.
25. Wolke, D., Bemerkungen über Eindeutigkeitsmengen additiver Funktionen. Elem. Math. 33(1) 1978, 1416.
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