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

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Perron's inversion formula

Statement

Let $F (s) = \sum_{n \geq 1} a_{n} n^{- s}$ converge absolutely on $ℜ s = c > 0$ . If its finite Dirichlet polynomials are dominated on the symmetric vertical segments by an integrable majorant permitting both indicated limits, then $A^{*} (x) = \frac{1}{2 π i} \int_{c - i \infty}^{c + i \infty} F (s) \frac{x^{s}}{s} d s (x > 0),$ where the integral is symmetric and $A^{*}$ is The starred summatory function.

Proof

Given: absolute convergence at $c$ and the stated domination hypothesis.

1.1givenalgebra

For $F_{N} (s) = \sum_{n \leq N} a_{n} n^{- s}$ , linearity and the kernel formula give $(2 π i)^{- 1} \int F_{N} (s) x^{s} d s / s = \sum_{n \leq N} a_{n} δ (x / n)$ .

2.1step 1.1given∎

Let first the height and then $N$ tend to infinity. The assumed domination permits both interchanges, while the right side tends exactly to the half-weighted sum defining $A^{*} (x)$ .

Depends on

Used by

Perron inversion for a finite Dirichlet polynomial Example

Dependency tree · one level

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Sources

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