Normal order of an arithmetic function

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

A function, perhaps simpler or better-understood, which "usually" takes the same or closely approximate values as a given arithmetic function.

Let $f$ be a function on the natural numbers. We say that the normal order of $f$ is $g$ if for every $\epsilon > 0$, the inequalities $$ (1-\epsilon) g(n) \le f(n) \le (1+\epsilon) g(n) $$ hold for almost all $n$: that is, 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 $\omega(n)$, the number of distinct prime factors of $n$, is $\log\log n$;
• The normal order of $\log d(n))$, where $d(n)$ is the number of divisors function of $n$, is $\log 2 \log\log n$.

See also

• Asymptotics of arithmetic functions
• Average order of an arithmetic function

References

• G.H. Hardy; S. Ramanujan; The normal number of prime factors of a number, Quart. J. Math., 48 (1917), pp. 76–92
• G.H. Hardy; E.M. Wright; An Introduction to the Theory of Numbers, Oxford University Press (2008), pp. 473. ISBN 0-19-921986-5
• Gérald Tenenbaum; Introduction to Analytic and Probabilistic Number Theory, ser. Cambridge studies in advanced mathematics 46 , Cambridge University Press (1995), pp. 299-324. ISBN 0-521-41261-7