Vol. 61, No. 1, 1975

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Generalized convolution ring of arithmetic functions

Ingrid Fotino

Vol. 61 (1975), No. 1, 103–116
Abstract

1. Introduction. The set of arithmetic functions has the structure of a unitary associative ring under functional addition and the convolution operation defined by

          ∑
(f∗g)(n) =    f (a)g(b) a,b,n ∈ N.
ab=n
(1.1)

It is also a unique factorization domain with respect to convolution.

The purpose of this paper is to determine the conditions under which this structure is preserved when the concept of convolution is generalized to include a weighting kernel γ:

          ∑
(fγ∗g)(n) =    f(a)g(b)γ(a,b)
ab=n
(1.2)

Mathematical Subject Classification 2000
Primary: 10A20, 10A20
Secondary: 20M05
Milestones
Received: 23 July 1974
Revised: 3 September 1975
Published: 1 November 1975
Authors
Ingrid Fotino