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Normal Order

Let typeset structure and typeset structure be two arithmetic functions. We say that typeset structure has the normal order typeset structure if for every positive typeset structure and almost all typeset structure we have

(1 - ϵ) F(n) < f(n) < (1 + ϵ) F(n) .(1)

The exceptional set  for which the  Equation 1 is false depends on typeset structure.

In other words, typeset structure has normal order typeset structure if there is a set of positive integers typeset structure of asymptotic density typeset structure such that

lim _ (n -> ∞,    n ∈ S) f(n)/F(n) = 1.(2)

Here the asymptotic density of a set typeset structure is typeset structure assuming the limit exists.

A function may possess a normal order, but no average order, or conversely. For instance, the function

f(n) = {1, if n is even, 2, if n is odd .

This function has the average order typeset structure, but no normal order. The function

f(n) = { m m a , if n = a , m 1, if n != a ,

for fixed typeset structure has normal order 1, but no average order.

Cite this web-page as:

Štefan Porubský: Normal Order.

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