TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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The convergence and absolute-convergence abscissae differ by at most one

Statement

For every Dirichlet series, its abscissae satisfy

$σ_{c} \leq σ_{a} \leq σ_{c} + 1$

in the extended real line.

Facts & Assumptions

Given: A Dirichlet series $D (s) = \sum_{n \geq 1} a_{n} n^{- s}$ with abscissae $σ_{c}$ and $σ_{a}$ .

[L1]

The two abscissae are defined by right-half-plane convergence and absolute convergence (The convergence and absolute-convergence abscissae of a Dirichlet series).

[L2]

Convergence at one point gives convergence on the entire open half-plane to its right (Convergence at one point of a Dirichlet series forces local uniform convergence on the open half-plane to its right).

Proof

technique · direct

1.1L1givenalgebra

Absolute convergence implies ordinary convergence term by term, so every half-plane counted for $σ_{a}$ is also counted for $σ_{c}$ . Therefore $σ_{c} \leq σ_{a}$ .

1.2L2givenalgebra

Let $s_{0}$ be any point of convergence and write $σ_{0} : = ℜ s_{0}$ . Then the terms $a_{n} n^{- s_{0}}$ tend to $0$ , so they are bounded: $∣ a_{n} ∣ \leq C n^{σ_{0}}$ for some $C$ . Hence for every $s$ with $ℜ s > σ_{0} + 1$ , $∣ a_{n} ∣ n^{- ℜ s} \leq C n^{- 1 - δ}$ for some $δ > 0$ , and the right-hand side is summable. Thus absolute convergence holds throughout $ℜ s > σ_{0} + 1$ .

2.1step 1.1step 1.2∎

Since step 1.2 applies at every point of convergence, taking infima in [L1] gives $σ_{a} \leq σ_{c} + 1$ . Combined with step 1.1, this is the claimed gap bound.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Leonard Tomczak, Analytic Number Theory, Theorem 3.4 (standard reference, not scraped)
Kiran S. Kedlaya, Notes on Analytic Number Theory, Lemma 2.2 (standard reference, not scraped)