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

Statement

Let $D (s) = \sum_{n \geq 1} a_{n} n^{- s}$ be a Dirichlet series. If it converges at some point $s_{0}$ , then it converges locally uniformly on the half-plane $ℜ s > ℜ s_{0}$ and therefore defines a holomorphic function there.

Facts & Assumptions

Given: A Dirichlet series $D (s) = \sum_{n \geq 1} a_{n} n^{- s}$ converging at $s_{0}$ , and a compact set $K \subseteq {s : ℜ s > ℜ s_{0}}$ .

[L1]

Abel summation for complex series rewrites $\sum_{n = M}^{N} u_{n} b_{n}$ in terms of the partial sums of $(u_{n})$ (Abel summation by parts for complex coefficients and their partial sums).

[L2]

A locally uniform limit of holomorphic functions is holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

Proof

technique · direct

1.1L1givenalgebra

Write $σ_{0} : = ℜ s_{0}$ and $u_{n} : = a_{n} n^{- s_{0}}$ . Since $\sum u_{n}$ converges, its partial sums $U_{N}$ are bounded: $∣ U_{N} ∣ \leq B$ . Put $ε : = \min_{s \in K} (ℜ s - σ_{0}) > 0.$ For $s \in K$ , apply [L1] to the tail with weights $b_{n} = n^{- (s - s_{0})}$ . Because $b_{n} - b_{n + 1} = O_{K} (n^{- 1 - ε})$ and $∣ b_{N} ∣ \leq N^{- ε}$ , there is a constant $C_{K}$ such that $∣ \sum_{n = M}^{N} a_{n} n^{- s} ∣ \leq C_{K} B (M^{- ε} + \sum_{n = M}^{N - 1} n^{- 1 - ε}) .$ The right-hand side tends to $0$ uniformly in $s \in K$ , so the series converges uniformly on $K$ .

2.1L2step 1.1∎

Each partial sum is holomorphic, being a finite linear combination of the holomorphic functions $s \mapsto n^{- s}$ . Since the convergence is uniform on every compact subset of the half-plane, [L2] makes the limit holomorphic there.

Depends on

Used by

Dependency tree · two levels

22 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

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