Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04
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The Riemann zeta function has no zeros when Re⁡s>1

Statement

If Re⁡s>1, then ζ(s)≠0.

Facts & Assumptions

Given: A complex number s with Re⁡s>1.

[L1]

On Re⁡s>1, ζ(s)=∏p(1−p−s)−1 (The Riemann zeta function has its Euler product on the half-plane Re⁡s>1).

[L2]

An absolutely convergent infinite product has nonzero value when every factor is nonzero (Absolute convergence criterion for complex infinite products).

Proof

technique · direct
1.1L1given

By [L1], ζ(s) is the reciprocal of the absolutely convergent product ∏p(1−p−s).

2.1givenstep 1.1L2algebra∎

Each factor 1−p−s is nonzero because ∣p−s∣=p−Re⁡s<1. By [L2], the absolutely convergent product is nonzero, so its reciprocal is also nonzero. Therefore ζ(s)≠0.

Depends on

Used by

Dependency tree · two levels

9 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