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Some Generalizations of Ramanujan's Sum

Published online by Cambridge University Press:  20 November 2018

K. G. Ramanathan  and
M. V. Subbarao
K. G. Ramanathan
Affiliation:
Tata Institute of Fundamental Research, Bombay, India
M. V. Subbarao
Affiliation:
University of Alberta, Edmonton, Alberta
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Ramanujan's well known trigonometrical sum C(m, n) denned by

where x runs through a reduced residue system (mod n), had been shown to occur in analytic problems concerning modular functions of one variable, by Poincaré [4]. Ramanujan, independently later, used these trigonometrical sums in his remarkable work on representation of integers as sums of squares [6]. There are various generalizations of C(m, n) in the literature (some also to algebraic number fields); see, for example, [9] which gives references to some of these. Perhaps the earliest generalization to algebraic number fields is due to H. Rademacher [5]. We here consider a novel generalization involving matrices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1250 - 1260
Copyright
Copyright © Canadian Mathematical Society 1980

References

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