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The K-Product of Arithmetic Functions

Published online by Cambridge University Press:  20 November 2018

A. A. Gioia
A. A. Gioia*
Affiliation:
University of Missouri and Texas Technological College
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In this note we introduce a natural generalization of the ordinary convolution of arithmetic functions: If f and g are arithmetic functions,

defines the K-product of f and g. If the kernel K(n) ≡ E(n) = 1, the K-product is the ordinary convolution Σd|nf(d)g(n/d);, if K(n) ≡ ∊(n) = [1/n], then the K-product is the unitary product Σf(d)g(n/d), summed over d|n, (d, n/d) = 1 (1, 2). We give in Theorem 1 a characterization of all associative kernels, i.e., kernels for which the corresponding K-product is associative.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 970 - 976
Copyright
Copyright © Canadian Mathematical Society 1965

References

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