DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

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Summatory functions and average orders

Definition

Let $f$ be an arithmetic function (Arithmetic functions on the positive integers). Its summatory function is

F_{f} (x) : = \sum_{n \leq x} f (n)

for real $x \geq 1$ .

An arithmetic function $g$ is an average order of $f$ when

\sum_{n \leq x} f (n) \sim \sum_{n \leq x} g (n)

as $x \to \infty$ , and the comparison sum on the right is eventually nonzero.

Remarks

This is a summatory asymptotic. It does not say that $f (n)$ and $g (n)$ are pointwise close term by term.
Because the index condition is $n \leq x$ , every summatory function here is constant on each interval $[m, m + 1)$ .

Depends on

Arithmetic functions on the positive integers

Used by

The average order of Euler's totient is 6n over pi squared Corollary
The average order of sigma is (pi squared over 6)n Corollary
The average order of tau is log n Corollary
The average order of the two-square representation count is pi Corollary
Dirichlet's hyperbola method for summatory convolutions Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)
Karl-Dieter Crisman, Number Theory: In Context and Interactive (standard reference, not scraped)