Normal order of an arithmetic function

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

$(1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) \,$

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

• G.H. Hardy; S. Ramanujan (1917). "The normal number of prime factors of a number". Quart. J. Math. 48: 76–92. http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper35/page1.htm. 
• G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. pp. 473. ISBN 0-19-921986-5. 
• Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. 46. Cambridge University Press. pp. 299–324. ISBN 0-521-41261-7. 

External links

• Weisstein, Eric W., "Normal Order" from MathWorld.

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