# Average order of an arithmetic function

From Encyclopedia of Mathematics

2020 Mathematics Subject Classification: *Primary:* 11A25 [MSN][ZBL]

Some simpler or better-understood function which takes the same values "on average" as an arithmetic function.

Let $f$, $g$ be functions on the natural numbers. We say that $f$ has average order $g$ if the asymptotic equality $$ \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) $$ holds as $x$ tends to infinity.

It is conventional to assume that the approximating function $g$ is continuous and monotone.

### Examples

- The average order of $d(n)$, the number of divisors of $n$, is $\log n$;
- The average order of $\sigma(n)$, the sum of divisors of $n$, is $ \frac{\pi^2}{6} n$;
- The average order of $\phi(n)$, the Euler totient function of $n$, is $ \frac{6}{\pi^2} n$;
- The average order of $r(n)$, the number of ways of expressing $n$ as a sum of two squares, is $\pi$;
- The Prime Number Theorem is equivalent to the statement that the von Mangoldt function $\Lambda(n)$ has average order 1.

### See also

### References

- G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
**ISBN**0-19-921986-5 - Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics
**46**. Cambridge University Press.**ISBN**0-521-41261-7

**How to Cite This Entry:**

Average order of an arithmetic function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Average_order_of_an_arithmetic_function&oldid=54520