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Mean Value theorems for multiplicative functions bounded in mean α-power, α >1

Published online by Cambridge University Press:  09 April 2009

P. D. T. A. Elliott
P. D. T. A. Elliott
Affiliation:
Univesity of Colorado, Boulder, Colorado, U.S.A.
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Abstract

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On analogy with functions if Lebesuge class Lα on the real line the author considers those multiplicative arthmetic functions which are bounded in mean α>1. Necessary and sufficient conditions are obtained in order that they should have a mean-value, zero or non-zero. An application is made to Ramanujan's τ-function.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 29 , Issue 2 , March 1980 , pp. 177 - 205
Copyright
Copyright © Australian Mathematical Society 1980

References

Atkinson, F. V. and Cherwell, Lord (1949), ‘The mean-values of arithmetical functions’, Quart. J. Math. Oxford Ser. 20, 65–79.CrossRefGoogle Scholar
Daboussi, H. and Delange, H. (1976), ‘On a theorem of P. D. T. A. Elliott on multiplicative functions’, J. London Math. Soc. 14, 345–356.CrossRefGoogle Scholar
Delange, H. (1961), ‘Un theéprème sur les fonctions arthmétiques multiplicatives et ses applications’, Ann. Sci. Éc. Norm. Sup., 3série 78 1–29.CrossRefGoogle Scholar
Dirichlet, P. G. L. (1949), ‘Über die Bestimmung der mittleren Werthe in der Zahlentheorie’, Abh. Akad. Berlin, 69–83.Google Scholar
Elliott, P. D. T. A. (1975), ‘A mean-value theorem for multiplicative functions’, Proc. Lond. Math. Soc. 3 31, 418–438.CrossRefGoogle Scholar
Elliott, P. D. T. A. (1980a), Probabilistic number theory (Springer, Heidelberg, New York).CrossRefGoogle Scholar
Elliott, P. D. T. A. (1980b), ‘High-power analogues of the Turán-Kubilius inequality, and an application to number theory’ Canad. J. Math., to appear.CrossRefGoogle Scholar
Erdös, P. (1946), ‘On the distribution function of additive functions,’ Ann. of Math. 47, 1–20.CrossRefGoogle Scholar
Gauss, C. F. (1801), Disquitiones arithmeticae, §5, Lipsiae, apud Gerh. Fleischer lun., 1801 = Werke, Band 1, Göttingen, 1870, 362–366.Google Scholar
Halász, G. (1968), ‘Über die Mitteiwerte multiplikativer zahlentheoretischer Funktionen’, Acta Math. Acad. Sci. Hungar. 19, 365–403.CrossRefGoogle Scholar
Hardy, G. H. (1927), ‘Note on Ramanujan's arithmetic function τ(n)’, Math. Proc. Cambridge Philos. Soc. 23, 675–680.CrossRefGoogle Scholar
Hardy, G. H. (1940), Ramanujan. Twelve lectures on subjects suggested by his life and work (Cambridge).Google Scholar
Hardy, G. H. (1949), Divergent series (Oxford).Google Scholar
Hardy, G. H. and Wright, E. M. (1960), An introduction to the theory of numbers, 4th ed. (Oxford).Google Scholar
Kubilius, J. (1962), Probabilistic methods in the theory of numbers, Amer. Math. Soc. Transi. of Math. Monographs no. 11 (Providence, 1964).Google Scholar
Levin, B. V. and Fainleib, A. S. (1967), ‘Applications of some integral equations to problems in number theory’ Uspekhi Mat. Nauk 22 (3) 135, 119–198 =Google Scholar
Russian Math. Surveys 22 (3) 05–06 1967, 119–204.CrossRefGoogle Scholar
Levin, B. V., Timofeev, N. M. and Tuliagonov, S. T. (1973), ‘The distribution of the values of multiplicative functions’, Liet. Mat. Rinkinys=Lit. Mat. Sb. 13 (1), 87–100.Google Scholar
Mordell, L. J. (1917), ‘On Mr. Ramanujan's empirical expansions of modular functions’, Math. Proc. Cambridge Philos. Soc. 19, 117–124.Google Scholar
Rankin, R. A. (1934), ‘Contributions to the theory of Ramanujan's function r(n) and similar arithmetical functions’, Math. Proc. Cambridge Philos. Soc. 35, 357–372.CrossRefGoogle Scholar
Ryavec, C. (1970), ‘A characterization of finitely distributed additive functions’, J. Number. Theory 2, 393–403.CrossRefGoogle Scholar
Wirsing, E. (1961), ‘Das asymptotische Verhalten von Summen uber multiplikative Funktionen’, Math. Ann. 143, 75–102.CrossRefGoogle Scholar
Wirsing, E. (1967), ‘Das asymptotische Verhalten von Summen über multiplikative Funktionen II’, Acta Math. Acad. Sci. Hungar. 18, 411–467.CrossRefGoogle Scholar