DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-31

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The power functions ${id}_{k}$ and the divisor-power-sum functions $σ_{k}$

Definition

For an integer $k$ , define the arithmetic function ${id}_{k}$ by

{id}_{k} (n) : = n^{k} .

The corresponding divisor-power-sum function is

σ_{k} (n) : = \sum_{\begin{array}{c} d ∣ n \\ d > 0 \end{array}} d^{k} .

In particular, $σ : = σ_{1}$ .

Remarks

Since $n > 0$ , the value $n^{k}$ is well defined for every integer $k$ and lies in the copy of $R \subset C$ from $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ .

Depends on

Used by

Euler's totient as the convolution μ*id₁ Example
The divisor functions arise by Dirichlet convolution Proposition

Dependency tree · two levels

13 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

Kiran S. Kedlaya, An Introduction to Analytic Number Theory, Definition 3.8 (standard reference, not scraped)
Victor Shoup, A Computational Introduction to Number Theory and Algebra, Exercise 2.50 (standard reference, not scraped)

The power functions id⁡kidk​ and the divisor-power-sum functions σkσk​