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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The symmetric Perron kernel

Statement

For $c, y > 0$ , define the integral by symmetric truncation. Then $\lim_{T \to \infty} \frac{1}{2 π i} \int_{c - i T}^{c + i T} \frac{y^{s}}{s} d s = {\begin{cases} 0 & 0 < y < 1, \\ 1 / 2 & y = 1, \\ 1 & y > 1. \end{cases}$

Proof

Given: $c, y > 0$ and the displayed symmetric truncations.

1.1

If $0 < y < 1$ , close the segment to the right by a semicircle and let its radius tend to infinity; $y^{s}$ decays there and no pole is enclosed, so the limit is $0$ . If $y > 1$ , close to the left instead; the enclosed simple pole at $0$ has residue $1$ , so the limit is $1$ .

givencases

2.1

If $y = 1$ , the integrand is $1 / s$ , and direct parametrisation gives $(2 π)^{- 1} \int_{- T}^{T} c (c^{2} + t^{2})^{- 1} d t \to 1 / 2$ . These three cases prove the claim.

step 1.1algebra∎

Used by

Perron's kernel at and away from its jump Example
The truncated Perron kernel Lemma
Perron's inversion formula Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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