Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T07:08:15.844Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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