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CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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Positive prime Dirichlet density does not give positive integer natural density

Statement refuted

A positive Dirichlet density among the primes does not imply a positive natural density inside all positive integers.

Facts & Assumptions

Given: The definitions of natural and Dirichlet density, the residue-class density theorem, and Chebyshev's upper bound for the prime-counting function (Natural and Dirichlet density, Primes in one reduced residue class have Dirichlet density 1 over phi(q), Chebyshev bounds for the prime-counting function).

Counterexample

technique · direct
1.1

Let P={p prime:p≡1(mod3)}. By Primes in one reduced residue class have Dirichlet density 1 over phi(q), P has relative Dirichlet density 1/2 among the primes.

givenalgebra
2.1

As a subset of the integers, however, its counting function is at most the prime-counting function π(x). By Chebyshev bounds for the prime-counting function, π(x)/x≤c2/log⁡x for all sufficiently large x, and this upper bound tends to 0. Hence P(x)/x→0, so P has natural density 0 among the positive integers despite its positive Dirichlet density among the primes.

step 1.1givenalgebra∎

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Dependency tree · two levels

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Sources