LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Zeta explicit formula zero free error balance

Statement

Assume countable choice.

For $x \geq 2$ and finite $T \geq 3$ , the classical region and truncated explicit formula give $ψ (x) - x = O (x e^{- c_{0} \log x / \log (T + 2)} \log^{2} T + \frac{x \log^{2} (x T)}{T} + \log x) .$ Constants may be enlarged and the positive region constant decreased. The zero sum used in the proof is finite.

Facts & Assumptions

Given: Countable choice and the data and hypotheses of the statement.

[F1]

The truncated von Mangoldt explicit formula: For $x, T \geq 2$ , $ψ_{0} (x) - x = - \sum_{∣ ℑ ρ ∣ < T} \frac{x^{ρ}}{ρ} - \frac{ζ^{'} (0)}{ζ (0)} - \frac{1}{2} \log (1 - x^{- 2}) + O ⁣ (\frac{x \log^{2} (x T)}{T} + (\log x) \min {1, \frac{x}{T ⟨ x ⟩}}),$ where $⟨ x ⟩$ is the distance to the nearest prime power other than possibly $x$ . The zero sum is finite and counts multiplicities.

[F2]

Riemann zeta classical zero free region: There is an absolute $c_{0} > 0$ such that $ζ$ has no zeros in $σ \geq 1 - c_{0} / \log (∣ t ∣ + 2)$ . The pole at $s = 1$ is not a zero.

[F3]

Zeta reciprocal zero sum bound: For $T \geq 2$ , the sum of $1 / ∣ ρ ∣$ over nontrivial zeros with $0 < ∣ Im ρ ∣ \leq T$ is $O (\log^{2} T)$ , with multiplicity. Adjoining any real nontrivial zeros preserves the estimate.

[F4]

The half-weighted Chebyshev function: For $x > 0$ , define $ψ_{0} (x) = \sum_{n < x} Λ (n) + {\begin{cases} Λ (x) / 2, & x \in Z_{> 0}, \\ 0, & x \notin Z_{> 0} . \end{cases}$ This differs at prime powers from the right-continuous $ψ (x) = \sum_{n \leq x} Λ (n)$ of def-chebyshev-psi-function.

Proof

1.1F2F3

Under countable choice, the zero-free region and reciprocal-zero bound apply. For each zero in the finite sum $∣ Im ρ ∣ < T$ , the region implies $∣ x^{ρ} ∣ \leq x \exp (- c_{0} \log x / \log (T + 2))$ . Summing absolute values and using the reciprocal estimate, including any real zeros, bounds the entire zero sum by the first displayed error.

2.1F1F4step 1.1∎

The supplied truncation error is at most $O (x \log^{2} (x T) / T + \log x)$ because its minimum is at most one. The fixed constant $ζ^{'} (0) / ζ (0)$ and $\log (1 - x^{- 2})$ are bounded for $x \geq 2$ . Finally $∣ ψ (x) - ψ_{0} (x) ∣ \leq (\log x) / 2$ , so replacing the half-weighted value gives the asserted error, including prime-power endpoints. The supplied formula holds for all x,T at these bounds; if a contour construction avoids ordinates, a non-ordinate in [T,T+1] has comparable bounds.

Depends on

Used by

Optimizing the prime number theorem contour height Example
Chebyshev psi prime number theorem error Theorem

Dependency tree · two levels

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Sources

§7.2, proof of Theorem 7.7 (standard reference, not scraped)