ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Numerics for the first and second Mertens theorems

Example

For $x = 10, 30, 100$ the weighted and reciprocal prime sums compare with their main terms as follows:

$\begin{array}{ccccc} x & \sum_{p \leq x} \frac{\log p}{p} & \log x & \sum_{p \leq x} \frac{1}{p} & \log \log x \\ 10 & 1.312652 & 2.302585 & 1.176190 & 0.834032 \\ 30 & 2.302016 & 3.401197 & 1.533439 & 1.224128 \\ 100 & 3.369471 & 4.605170 & 1.802817 & 1.527180 \end{array}$

Facts & Assumptions

Given: The cutoffs $x = 10, 30, 100$ .

[L1]

The first Mertens theorem controls $\sum_{p \leq x} (\log p) / p$ by $\log x$ (Mertens' first theorem for primes).

[L2]

The second Mertens theorem controls $\sum_{p \leq x} 1 / p$ by $\log \log x$ (Mertens' second theorem for primes).

Verification

technique · direct

1.1L1L2givenalgebra

Summing over the primes up to $10$ , $30$ , and $100$ gives the four numerical columns in the displayed table.

2.1L1L2step 1.1∎

At each of these three cutoffs, the reciprocal sum is numerically closer to $\log \log x$ than the weighted sum is to $\log x$ . This small-range comparison is consistent with the nonzero bounded terms allowed by [L1] and [L2], but it is numerical evidence only and does not compare their asymptotic error strengths.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Leo Goldmakher, A Quick Proof of Mertens' Theorem (standard reference, not scraped)