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Let and
be two arithmetic functions. We say that
has the normal order
if for every positive
and almost all
we have
![]() | (1) |
The exceptional set for which the Equation 1 is false depends on .
In other words, has normal order
if there is a set of positive integers
of asymptotic density
such that
![]() | (2) |
Here the asymptotic density of a set is
assuming the limit exists.
A function may possess a normal order, but no average order, or conversely. For instance, the function
This function has the average order , but no normal order. The function
for fixed has normal order 1, but no average order.
Cite this web-page as:
Štefan Porubský: Normal Order.