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Keywords:
generalized arithmetical function; semimultiplicative function; Selberg multiplicative function; Dirichlet convolution
generalized arithmetical function; semimultiplicative function; Selberg multiplicative function; Dirichlet convolution
Summary:
By a generalized arithmetical function we mean a function from the set of positive integers to a ring with identity, and we say that a generalized arithmetical function $f$ is semimultiplicative if $f(n) = c_f f_M(n/a_f)$, where $c_f$ is a unit in the ring, $a_f$ is a positive integer and $f_M$ is a multiplicative generalized arithmetical function. We study basic properties of these functions, connections to Selberg multiplicative functions and to the Dirichlet convolution. Particular attention is paid to the commutativity and noncommutativity of the function values.
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