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 $ζ (s) \neq 0$ .

Facts & Assumptions

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

[L1]

On $Re s > 1$ , $ζ (s) = \prod_{p} (1 - p^{- s})^{- 1}$ (The Riemann zeta function has its Euler product on the half-plane $Re s > 1$ ).

[L2]

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

Proof

technique · direct

1.1

By [L1], $ζ (s)$ is the reciprocal of the absolutely convergent product $\prod_{p} (1 - p^{- s})$ .

L1given

2.1

By [L2], that product is nonzero, so its reciprocal is also nonzero. Therefore $ζ (s) \neq 0$ .

step 1.1L2algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 6 §2 (standard reference, not scraped)

The Riemann zeta function has no zeros when Re⁡s>1Res>1