Open Research Index

teal-sea/zeta-lab

Thomas Lince

Palomar Registrypalomar/PALOMAR-2026-08-21-000012Aug 21, 2026math.NTscore 0

Davenport and Heilbronn (1936) exhibited a Dirichlet series with real coefficients satisfying a Riemann-type functional equation which nevertheless has zeros off the critical line. It is the standard demonstration that a functional equation of Riemann type does not on its own force the Riemann Hypothesis, it is the reference counterexample against which structural explanations of the Riemann Hypothesis are tested, and the location of its zeros remains an active subject. The function is DH(s) = (1 - i*kappa)/2 * L(s, chi) + (1 + i*kappa)/2 * L(s, chi^-1) for a quartic Dirichlet character chi…

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