ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A Dirichlet-convolution table through $12$

Example

For the constant-one function $1$ , the convolution $(1 * 1) (n)$ counts positive divisors. Through $12$ one gets:

$n$	positive divisors of $n$	$(1 * 1) (n)$	$τ (n)$
$1$	$1$	$1$	$1$
$2$	$1, 2$	$2$	$2$
$3$	$1, 3$	$2$	$2$
$4$	$1, 2, 4$	$3$	$3$
$5$	$1, 5$	$2$	$2$
$6$	$1, 2, 3, 6$	$4$	$4$
$7$	$1, 7$	$2$	$2$
$8$	$1, 2, 4, 8$	$4$	$4$
$9$	$1, 3, 9$	$3$	$3$
$10$	$1, 2, 5, 10$	$4$	$4$
$11$	$1, 11$	$2$	$2$
$12$	$1, 2, 3, 4, 6, 12$	$6$	$6$

Facts & Assumptions

Given: The constant-one function $1$ and the integers $1 \leq n \leq 12$ .

Verification

technique · direct

1.1

By Dirichlet convolution of arithmetic functions, $(1 * 1) (n) = \sum_{d ∣ n} 1$ , so each entry in the third column is exactly the number of listed positive divisors of $n$ . The rows $n = 1, \dots, 6$ therefore give the values $1, 2, 2, 3, 2, 4$ .

givenalgebra

1.2

The same divisor-counting rule applied to the listed divisors for $n = 7, \dots, 12$ gives the remaining values $2, 4, 3, 4, 2, 6$ . By The divisor-counting function $τ$ , these are also the corresponding $τ (n)$ values.

givenalgebra

2.1

Thus every row of the table displays $(1 * 1) (n) = τ (n)$ on the range $1 \leq n \leq 12$ .

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Section 2.9 (standard reference, not scraped)

A Dirichlet-convolution table through 1212