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Some Generalizations of an Identity of Subhankulov

Published online by Cambridge University Press:  20 November 2018

D. Suryanarayana  and
David T. Walker
D. Suryanarayana
Affiliation:
Department of Mathematical Sciences Memphis State University, Memphis, Tennessee 38152
David T. Walker
Affiliation:
Department of Mathematical Sciences Memphis State University, Memphis, Tennessee 38152
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Abstract

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In 1957, M. A. Subhankulov established the following identity

where ; μ is the Môbius function and J2 is the Jordan totient function of order 2. Since the Ramanujan trigonometrical sum C(nr) = ∑d| (n, r)dμ(r/d), we rewrite the above identity using C(n, r).

In this paper, we give a generalization of Ramanujan's sum, which generalizes some of the earlier generalizations mainly due to E. Cohen, and prove a theorem from which we deduce some generalizations of the above identity.

Keywords

10A2010A99Möbius functionJordan totient functiongeneralized Ramanujan's sumk-free integersRiemann zeta function
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 4 , 01 December 1977 , pp. 489 - 494
Copyright
Copyright © Canadian Mathematical Society 1977

References

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