John R. Stallings

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

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

$(1-\epsilon )g(n)\le f(n)\le (1+\epsilon )g(n)$

hold for almost all n: that is, if the proportion of n ≤ x 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

• The Hardy–Ramanujan theorem: the normal order of ω(n), the number of distinct prime factors of n, is log(log(n));
• The normal order of Ω(n), the number of prime factors of n counted with multiplicity, is log(log(n));
• The normal order of log(d(n)), where d(n) is the number of divisors of n, is log(2) log(log(n)).

See also

• Average order of an arithmetic function
• Divisor function
• Extremal orders of an arithmetic function

References

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