Average order of an arithmetic function

Average order of an arithmetic function

In number theory, the average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".

Let "f" be an arithmetic function. We say that the "average order" of "f" is "g" if

: sum_{n le x} f(n) sim sum_{n le x} g(n)

as "x" tends to infinity.

It is conventional to choose an approximating function "g" that is continuous and monotone.

Examples

* The average order of "d"("n"), the number of divisors of "n", is log("n");
* The average order of σ("n"), the sum of divisors of "n", is π2 / 6;
* The average order of φ("n"), Euler's totient function of "n", is 6 / π2;
* The average order of "r"("n"), the number of ways of expressing "n" as a sum of two squares, is π;
* The average order of ω("n"), the number of distinct prime factors of "n", is log log "n";
* The average order of Ω("n"), the number of prime factors of "n", is log log "n";
* The prime number theorem is equivalent to the statement that the von Mangoldt function Λ("n") has average order 1.

ee also

* Divisor summatory function
* Normal order of an arithmetic function

References

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