Average order of an arithmetic function

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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.


See also


  • 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
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