Average order of an arithmetic function

In number theory, an 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 an average order of f is g if

${\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}$

as x tends to infinity.

It is conventional to choose an approximating function g that is continuous and monotone. But even thus an average order is of course not unique.

Examples

• An average order of d(n), the number of divisors of n, is log(n);
• An average order of σ(n), the sum of divisors of n, is nπ² / 6;
• An average order of φ(n), Euler's totient function of n, is 6n / π²;
• An average order of r(n), the number of ways of expressing n as a sum of two squares, is π;
• An average order of ω(n), the number of distinct prime factors of n, is log log n;
• An 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;
• An average order of μ(n), the Möbius function, is zero; this is again equivalent to the prime number theorem.

Better average order

This notion is best discussed through an example. From

${\displaystyle \sum _{n\leq x}d(n)=x\log x+(2\gamma -1)x+o(x)}$ 

(${\displaystyle \gamma }$  is the Euler-Mascheroni constant) and

${\displaystyle \sum _{n\leq x}\log n=x\log x-x+O(\log x),}$ 

we have the asymptotic relation

${\displaystyle \sum _{n\leq x}(d(n)-(\log n+2\gamma ))=o(x)\quad (x\rightarrow \infty ),}$ 

which suggests that the function ${\displaystyle \log n+2\gamma }$  is a better choice of average order for ${\displaystyle d(n)}$  than simply ${\displaystyle \log n}$ .

See also

• Divisor summatory function
• Normal order of an arithmetic function
• Extremal orders of an arithmetic function

References

• Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press. ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001. Pp.347–360
• Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press. pp. 36–55. ISBN 0-521-41261-7. Zbl 0831.11001.