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

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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

1.1givenalgebra

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 $\sum_{n \leq 50} τ (n) = 207$ .

2.1step 1.1givenalgebra

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

3.1step 2.1given∎

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).

Depends on

The summatory divisor-counting function is x log x plus (2 gamma - 1)x plus O(sqrt x)

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Karl-Dieter Crisman, Number Theory: In Context and Interactive (standard reference, not scraped)