LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Monotone chebyshev tauberian desmoothing

Statement

Let $A : [1, \infty) \to [0, \infty)$ be nondecreasing and locally integrable, with $A (x) = O (x)$ , and let $a \geq 0$ . If $\int_{1}^{\infty} (A (x) - a x) x^{- 2} d x$ converges, then $A (x) / x \to a$ .

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Cauchy criterion for improper integrals: The integral $\int_{a}^{\infty} f$ converges if and only if, for every $ε > 0$ , there is $A > a$ such that $A \leq u < v ⟹ ∣ \int_{u}^{v} f ∣ < ε .$ At a finite right singular endpoint $b$ , replace the condition by $b - δ < u < v < b$ ; at a finite left endpoint use $a < u < v < a + δ$ ; at $- \infty$ use $u < v \leq - A$ . In each case all displayed proper integrals must exist.

Proof

1.1

Fix $λ > 1$ . The Cauchy criterion makes both tail integrals over $[x, λ x]$ and $[x / λ, x]$ tend to zero. Monotonicity gives $o (1) \geq \frac{A (x)}{x} (1 - λ^{- 1}) - a \log λ, o (1) \leq \frac{A (x)}{x} (λ - 1) - a \log λ .$ These inequalities hold for sufficiently large x that $x / λ \geq 1$ .

F1given

2.1

Therefore $lim sup A (x) / x \leq a λ \log λ / (λ - 1)$ and $lim inf A (x) / x \geq a \log λ / (λ - 1)$ . Let $λ ↓ 1$ ; both constants tend to a. This also works when a=0 (and nonnegativity supplies a zero lower bound). No differentiation of A or continuity at its jumps was used.

step 1.1algebra∎

Depends on

Cauchy criterion for improper integrals

Used by

Newman tauberian prime number theorem Example
Prime number theorem arithmetic progressions Theorem

Dependency tree · two levels

12 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

§1.3, monotonicity argument preceding §1.4 (standard reference, not scraped)