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Multiplicative functions and Ramanujan's τ-function

Published online by Cambridge University Press:  09 April 2009

P. D. T. A. Elliott
P. D. T. A. Elliott
Affiliation:
Imperial College, London
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Abstract

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It is proved that (|τ(n)|n−11/2)δ has a mean-value for 0 <δ > < 2, where τ(n) is Ramanujan's function from modular arithmetic. Some further results are conjectured.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 4 , April 1981 , pp. 461 - 468
Copyright
Copyright © Australian Mathematical Society 1981

References

Elliott, P. D. T. A. (1979), Probabilistic number theory, volume I (Grundlehren Series number 239, Springer, Berlin, New York).CrossRefGoogle Scholar
Elliott, P. D. T. A. (1980a), Probabilistic number theory, volume II (Grundlehren Series number 240, Springer, Berlin. New York).CrossRefGoogle Scholar
Elliott, P. D. T. A. (1980b), ‘Mean value theorems for multiplicative functions bounded in mean α-power, α < 1’, J. Austral. Math. Soc. Ser. A 29, 177–205.CrossRefGoogle Scholar
Hardy, G. H. (1949), Divergent series, Oxford.Google Scholar
Kubilius, J. (1964), Probabilistic methods in the theory of numbers, Amer. Math. Soc. Translations of Math. Monographs, No. 11, Providence.CrossRefGoogle 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 τ(n) and similar arithmetical functions’, Math. Proc. Cambridge Philos. Soc. 35, 357–372.CrossRefGoogle Scholar