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# Average order of an arithmetic function: Wikis

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# Encyclopedia

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.

## References

• G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed. ed.). Oxford University Press. pp. 347–360. 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. 36–55. ISBN 0-521-41261-7.