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

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The truncated Perron kernel

Statement

Let $I (y, T) = (2 π i)^{- 1} \int_{c - i T}^{c + i T} y^{s} d s / s$ and let $δ (y)$ be the three-valued kernel of The symmetric Perron kernel. For $c, y, T > 0$ , $∣ I (y, T) - δ (y) ∣ < {\begin{cases} y^{c} \min {1, (T ∣ \log y ∣)^{- 1}}, & y \neq 1, \\ c / T, & y = 1. \end{cases}$

Proof

Given: $c, y, T > 0$ and the symmetric kernel value $δ (y)$ .

1.1givencases

For $0 < y < 1$ , move the finite segment rightward and then let the new real part tend to infinity; its two horizontal tails have modulus at most $y^{c} / (T ∣ \log y ∣)$ . A circular arc gives the independent bound $y^{c}$ . The leftward contours give the same two bounds for $y > 1$ , after subtracting the residue $1$ .

2.1step 1.1algebra∎

For $y = 1$ , integration of $d s / s$ on the two omitted tails gives $∣ I (1, T) - 1 / 2 ∣ < c / T$ . Taking the smaller of the two preceding bounds proves the stated estimate.

Depends on

The symmetric Perron kernel

Used by

Perron's kernel at and away from its jump Example
A truncated Perron formula Theorem

Dependency tree · one level

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Sources

Kiran S. Kedlaya, Analytic Number Theory, Lemma 10.2 (standard reference, not scraped)