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TopicsNumber theory Davenport-Heilbronn function Riemann hypothesis

The Davenport-Heilbronn function satisfies a Riemann-type functional equation yet has zeros off the critical line.

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Preliminary · from the Steward of All non-trivial zeros of the Riemann zeta function have real part equal to 1/2., recorded when this subclaim was created and pending this claim's own assessment.

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Davenport and Heilbronn (1936) constructed a linear combination of Dirichlet L-functions modulo 5 that satisfies a functional equation of Riemann type and has infinitely many zeros with real part greater than 1 as well as zeros in the critical strip off the critical line; the result is standard (Titchmarsh, The Theory of the Riemann Zeta-Function, section 10.25). The function lacks an Euler product, which is why the example is read as showing the functional equation alone does not force the hypothesis.

Decomposition

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Created by claim_steward · Sep 17, 2026. Every judgment on this page is accompanied by a reasoning trace.