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The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane $Re s > 1$

Statement

For every complex number $s$ with $Re s > 1$ , the series

$\sum_{n = 1}^{\infty} n^{- s}, n^{- s} : = \exp (- s \log n),$

converges absolutely. Moreover, if $K \subseteq {s \in C : Re s > 1}$ is compact, then the same series converges uniformly on $K$ .

Facts & Assumptions

Given: A compact set $K \subseteq {s \in C : Re s > 1}$ .

[L1]

The complex exponential is $\exp z = \sum_{m \geq 0} z^{m} / m!$ (The complex exponential by its power series).

[L2]

The real logarithm is defined on $(0, \infty)$ , so $\log n$ is defined for every integer $n \geq 1$ (The natural logarithm as the inverse of the exponential function).

[L3]

If $∣ f_{n} (s) ∣ \leq M_{n}$ on a set and $\sum M_{n}$ converges, then $\sum f_{n}$ converges uniformly there (Weierstrass M-test for complex-valued function series).

[L4]

For rational $p > 1$ , the series $\sum_{n \geq 1} n^{- p}$ converges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ).

Proof

technique · direct

1.1

Because $K$ is compact and lies in the open half-plane $Re s > 1$ , there is a real number $σ > 1$ with $Re s \geq σ$ for every $s \in K$ . For $n \geq 1$ and $s \in K$ , [L1] and [L2] give $n^{- s} = \exp (- s \log n)$ , so $∣ n^{- s} ∣ = \exp (- Re s \log n) \leq \exp (- σ \log n) = n^{- σ} .$

givenL1L2choosealgebra

2.1

Taking $σ > 1$ rational if necessary, [L4] makes $\sum_{n \geq 1} n^{- σ}$ convergent. Step 1.1 and [L3] therefore give uniform convergence of $\sum n^{- s}$ on $K$ . Since $∣ n^{- s} ∣$ is bounded by the same summable majorant, the series also converges absolutely at each point of $K$ , hence at each $s$ with $Re s > 1$ .

step 1.1L3L4algebra∎

Depends on

Used by

The Riemann zeta function has its Euler product on the half-plane Res>1 Theorem

Cited to discharge well-definedness by The Riemann zeta function on the half-plane Res>1.

Dependency tree · two levels

28 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

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 6 §2 (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 11 §3 (standard reference, not scraped)

The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane Re⁡s>1Res>1