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 such that an identity of the form

*n*, where is the characteristic function of the set of

*k*-free integers and 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 and .

**[1]**L. Carlitz and M. V. Subbarao,*On a class of multiplicative functions*, Duke Math. J. (to appear).**[2]**Eckford Cohen,*An extension of Ramanujan’s sum*, Duke Math. J.**16**(1949), 85–90. MR**0027781****[3]**Eckford Cohen,*An elementary estimate for the 𝑘-free integers*, Bull. Amer. Math. Soc.**69**(1963), 762–765. MR**0153628**, 10.1090/S0002-9904-1963-11024-1**[4]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 4th ed., Oxford Univ. Press, London, 1960.

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0292778-8

Keywords:
*k*-free integers,
generalized Ramanujan sum,
Riemann zeta function

Article copyright:
© Copyright 1972
American Mathematical Society