Terence Tao sketches a proof excluding Siegel zeroes

Terence Tao sketches a proof excluding Siegel zeroes

Terence Tao sketches Ben Green’s formulation of a proof that excludes Siegel zeroes, zeros of Dirichlet L-functions implicated in ineffective error bounds for prime-counting results. Tao contrasts this largely algebraic-number-theory argument with other recent approaches based on analysis, which establish a stronger “quasi-Riemann hypothesis.” The proof seeks to bound how often a Legendre symbol is negative at primes in a specified range. Its central construction turns that counting problem into a question about an integer: Tao builds a lattice from polynomials evaluated, along with their horizontal derivatives, on a carefully chosen grid. The lattice’s squared covolume is a non-zero integer, so its prime divisors and its size can be bounded in two different ways. The key move is to use primes for which the grid lies p-adically close to a curve. That closeness gives a lower bound on the covolume’s divisibility by those primes; an Archimedean bound controls its overall size. Together, the bounds yield the desired estimate. Tao says multiplicity is an innovation in Green’s treatment, and identifies the polynomial method as the parity-breaking input that lets this proof avoid classical analytic number theory methods. The captured evidence omits the mathematical formulas, so the precise parameter choices and numerical bounds cannot be assessed here.

Publication: 01843319-a483-4e77-8117-b9e1d126fe90

attribution — pass: The reading correctly says Terence Tao is sketching Ben Green's formulation of the argument. It also keeps the parity-breaking and multiplicity remarks attributed to Tao.

I would like here to present a sketch of Green’s formulation (with some small notational changes), which I find particularly elegant.

summary — uncertain: The reading leaves out the central AI context. Tao says the original nine-page paper and the quasi-Riemann hypothesis papers came from an OpenAI release, and he discloses using the SAIR open math model. Without that, a general reader cannot see why this matters here. Central terms (Legendre symbol, lattice, covolume, p-adic, Archimedean, parity problem) are also left unexplained for non-specialists. Smallest repair: add the OpenAI origin and Tao's AI disclosure, and add tooltips for these terms.

However, there was a third paper in the same release that did not establish the quasi-Riemann hypothesis, but instead excluded Siegel zeroes directly. This paper was significantly shorter (nine pages)

date — pass: originalPublished (2026-10-19T18:27:19Z, from the publisher page) matches the post's URL date and the feed entry. No earlier first publication is indicated.

support — uncertain: Claim 5 says previous treatments required an 'extension of the grid.' The captured text drops the symbol after 'extension of', so the source does not show what was extended; it may be a ring or field rather than the grid. Smallest repair: say 'a (somewhat artificial) extension' without naming the grid. The other claims are grounded. The question's phrase 'Legendre symbol is a quadratic non-residue' slightly misstates the source, which treats the number as the non-residue, not the symbol.

previous treatments did not do this, and had to work with a (somewhat artificial) extension of , such as , in order to get comparably non-trivial bounds.

media — pass: The selected media is the verified direct link to Tao's original post, which is the most relevant and accessible source.