TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The truncated von Mangoldt explicit formula

Statement

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.

Proof

Given: $x, T \geq 2$ .

1.1

Apply truncated Perron inversion to $- ζ^{'} / ζ$ . Its kernel error is bounded by separating the nearest other prime power from the remaining terms, giving the stated Perron error.

givenalgebra

2.1

In a finite rational grid within bounded distance of $T$ , the unit-interval zero bound leaves a height at distance $≫ 1 / \log T$ from every zero ordinate. Shift at that height using the two logarithmic-derivative bounds and the residue ledger; changing back to $T$ affects only $O (\log T)$ finite zero terms and is absorbed in the displayed error.

step 1.1construct∎

Depends on

Used by

An unordered infinite zero sum is not an explicit formula Counterexample
Smoothed versus sharp explicit formulas Example

Dependency tree · two levels

22 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

Kiran S. Kedlaya, Analytic Number Theory, Theorem 10.1 (standard reference, not scraped)