Normal order of an arithmetic function

From Citizendium
Jump to navigation Jump to search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, in the field of number theory, the normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values.

Let f be a function on the natural numbers. We say that the normal order of f is g if for every ε > 0, the inequalities

hold for almost all n: that is, if the proportion of nx for which this does not hold tends to 0 as x tends to infinity.

It is conventional to assume that the approximating function g is continuous and monotone.

Examples

See also

References

  • G.H. Hardy; S. Ramanujan (1917). "The normal number of prime factors of a number". Quart. J. Math. 48: 76–92.