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Average Orders Divisor Sums and Representation Counts — Examples

1 · Prerequisites

2 · Summary

These examples keep the two main computations concrete. The first draws one small hyperbola split exactly, and the second shows what the divisor-summatory error looks like numerically before any asymptotic language hides the table.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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

A small lattice decomposition for Dirichlet's hyperbola method

Example

Take $x = 12$ , $U = 3$ , and $V = 4$ , so $U V = 12$ . For $τ = 1 * 1$ , Dirichlet's hyperbola identity becomes

\sum_{n \leq 12} τ (n) = \sum_{a \leq 3} ⌊ \frac{12}{a} ⌋ + \sum_{b \leq 4} ⌊ \frac{12}{b} ⌋ - 3 \cdot 4.

Numerically this is

35 = (12 + 6 + 4) + (12 + 6 + 4 + 3) - 12.

Facts & Assumptions

Given: The value $x = 12$ and the split $U = 3$ , $V = 4$ .

Verification

technique · direct

1.1

By The divisor functions arise by Dirichlet convolution, $τ = 1 * 1$ . Therefore Dirichlet's hyperbola method for summatory convolutions applies with $F (y) = G (y) = ⌊ y ⌋$ .

given

2.1

The two arm sums are $\sum_{a \leq 3} ⌊ \frac{12}{a} ⌋ = 12 + 6 + 4 = 22, \sum_{b \leq 4} ⌊ \frac{12}{b} ⌋ = 12 + 6 + 4 + 3 = 25,$ and the overlap is $F (3) G (4) = 3 \cdot 4 = 12$ .

step 1.1givenalgebra

3.1

Hence the hyperbola identity gives $\sum_{n \leq 12} τ (n) = 22 + 25 - 12 = 35$ . This matches the direct total $1 + 2 + 2 + 3 + 2 + 4 + 2 + 4 + 3 + 4 + 2 + 6 = 35$ .

step 2.1givenalgebra∎

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

The divisor summatory estimate through several small values

Example

Let

E (x) : = \sum_{n \leq x} τ (n) - x \log x - (2 γ - 1) x .

Using $γ \approx 0.5772156649$ , the theorem predicts $E (x) = O (\sqrt{x})$ . For four small values one gets:

$x$	$\sum_{n \leq x} τ (n)$	$x \log x + (2 γ - 1) x$	$E (x)$	$E (x) / \sqrt{x}$
$5$	$10$	$8.819346$	$1.180654$	$0.528004$
$10$	$27$	$24.570164$	$2.429836$	$0.768382$
$20$	$66$	$63.003272$	$2.996728$	$0.670089$
$50$	$207$	$203.322717$	$3.677283$	$0.520046$

Facts & Assumptions

Given: The definition of $E (x)$ above.

Verification

technique · direct

1.1

The exact divisor sums are obtained by direct counting: $\sum_{n \leq 5} τ (n) = 10$ , $\sum_{n \leq 10} τ (n) = 27$ , $\sum_{n \leq 20} τ (n) = 66$ , and $\sum_{n \leq 50} τ (n) = 207$ .

givenalgebra

2.1

Substituting these four values into the displayed definition of $E (x)$ produces the residual column and then the scaled residual column $E (x) / \sqrt{x}$ .

step 1.1givenalgebra

3.1

The quotients in the last column stay on the scale of a bounded constant rather than growing like a positive power of $x$ , which is exactly the $\sqrt{x}$ scale asserted by The summatory divisor-counting function is x log x plus (2 gamma - 1)x plus O(sqrt x).

step 2.1given∎

Average Orders Divisor Sums and Representation Counts — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A small lattice decomposition for Dirichlet's hyperbola method

Example

Facts & Assumptions

Verification

The divisor summatory estimate through several small values

Example

Facts & Assumptions

Verification

Sources