Minerval

← the claim it bears on

A line of reasoning from premises to a conclusion. Like a claim, it belongs to no one who makes it.constitution →

Skeptical considerations

This argument, if it holds, weighs against the claim.constitution →All non-trivial zeros of the Riemann zeta function have real part equal to 1/2.

The argument

Because the Davenport–Heilbronn function satisfies a Riemann-type functional equation yet has zeros off the critical line, the functional equation alone cannot force the zeros onto the line, so any argument for the hypothesis must use the Euler product in an essential way; and because the Mertens conjecture, which had extensive numerical support and would have implied the hypothesis, was disproved by Odlyzko and te Riele in 1985, while the computed zeros form a vanishingly small initial segment in which the Lehmer phenomenon of near-coincident zeros already appears, some analysts hold that the numerical evidence is weaker than it looks and that the hypothesis could fail at heights beyond computational reach.

Does it work?

Not yet evaluated.

Premises

Where it is made

Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.

No source is recorded as making it.

Earlier ratings

  • Sep 17, 2026

    Its factual premises are settled: the Davenport–Heilbronn function has zeros off the line and the Mertens conjecture is false. What they establish is limited: that the functional equation alone cannot prove the hypothesis and that numerical evidence in this area is less conclusive than it appears. They are reasons to withhold certainty, not evidence that any zeta zero lies off the line, and Farmer's reply argues that random-matrix models account for each phenomenon without threatening the hypothesis.