A pivotal point for certain
problems in probabilistic number theory is that there exists a positive constant c such
that for every member f of the family of additive complex valued arithmetic
functions
where
and
pα being a power of a prime number. This paper considers the extension of this
property in two directions suggested by Harold N. Shapiro. First, an investigation is
made of when this property holds for weight functions other than w(m) ≡ 1. Second,
it is shown that this property can be extended to various nonadditive arithmetic
functions.
