On an identity of Eckford Cohen
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- by M. V. Subbarao and D. Suryanarayana PDF
- Proc. Amer. Math. Soc. 33 (1972), 20-24 Request permission
Abstract:
We characterize all multiplicative arithmetical functions ${f_k}(r)$ such that an identity of the form \[ \sum \limits _{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,} \] 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 $g(k) = \zeta (k)$.References
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L. Carlitz and M. V. Subbarao, On a class of multiplicative functions, Duke Math. J. (to appear).
- Eckford Cohen, An extension of Ramanujan’s sum, Duke Math. J. 16 (1949), 85–90. MR 27781
- Eckford Cohen, An elementary estimate for the $k$-free integers, Bull. Amer. Math. Soc. 69 (1963), 762–765. MR 153628, DOI 10.1090/S0002-9904-1963-11024-1 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960.
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 33 (1972), 20-24
- MSC: Primary 10H99
- DOI: https://doi.org/10.1090/S0002-9939-1972-0292778-8
- MathSciNet review: 0292778