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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On an identity of Eckford Cohen

Authors: M. V. Subbarao and D. Suryanarayana
Journal: Proc. Amer. Math. Soc. 33 (1972), 20-24
MSC: Primary 10H99
MathSciNet review: 0292778
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Abstract: We characterize all multiplicative arithmetical functions $ {f_k}(r)$ such that an identity of the form

$\displaystyle \sum\limits_{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,} $

holds for all n, where $ {q_k}(n)$ is the characteristic function of the set of k-free integers and $ {c_k}(n,r)$ is the generalized Ramanujan sum. This characterization yields several arithmetical identities of the above form including an identity of Eckford Cohen, which occurs as a special case of our theorem on taking $ {f_k}(r) = \mu (r)/{J_k}(r)$ and $ g(k) = \zeta (k)$.

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Keywords: k-free integers, generalized Ramanujan sum, Riemann zeta function
Article copyright: © Copyright 1972 American Mathematical Society