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CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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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.

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Two-sided Chebyshev bounds do not imply the prime number theorem

Statement refuted

The existence of positive constants c1<c2 with

c1xlog⁡x≤f(x)≤c2xlog⁡x

for all sufficiently large x forces

f(x)∼xlog⁡x.

Facts & Assumptions

Given: The refuted implication and the Chebyshev bound shape from Chebyshev bounds for the prime-counting function.

[L1]

For x>e, one has log⁡x>1>0, so the quotient x/log⁡x is defined.

Counterexample

technique · direct
1.1L1givenconstructalgebra

Let f(x):=32xlog⁡x(x>e). By [L1] this is well defined, and 1⋅xlog⁡x≤f(x)≤2⋅xlog⁡x for every x>e, so the two-sided Chebyshev-type bounds hold.

2.1step 1.1algebra∎

But f(x)x/log⁡x=32 for every x>e, so the ratio does not tend to 1. Therefore the displayed implication is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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