Natalie Trono

Member of Non-Technical Staff @PlasmaAI

[Natalie Trono @Natalie_Trono]

@horwitzben Would buy

[Natalie Trono @Natalie_Trono]

@sama @PlasmaAI

[Plasma AI @PlasmaAI]

@OpenAI recently released a proof of the quasi-Riemann Hypothesis, which says there is some fixed θ < 1 such that no zero of the Riemann zeta function has real part greater than θ. They prove it with θ = 7/8. The paper hasn't been peer reviewed yet, but its main theorem has a computer-checked proof in Lean.

This improves the bound on the root-mean-square error of the prime count against li(N). In general, that error is bounded by roughly N^θ if and only if no zero has real part greater than θ. The Riemann Hypothesis is the case θ = 1/2. So the best proven bound has dropped from almost N to roughly N^(7/8).

That's real progress, but the bound is still far above a random set's √(N/ln(N)) error, by a factor of about N^(3/8). It's worth noting that proving that the primes are merely as steady as random is sufficient to prove the full Riemann Hypothesis.

[Nick Diao @nicholas_diao]

Our team built on @OpenAI's recent mathematical breakthroughs to resolve Erdős problem 408. Exciting to see how quickly progress in this field is compounding.

[Natalie Trono @Natalie_Trono]

@PlasmaAI @OpenAI 🕺

[Plasma AI @PlasmaAI]

More than 50 years after Erdős posed it, we have resolved Erdős problem 408: for almost all n, totient iteration takes ∼α log n steps to reach 1, for some constant α. The proof builds on @OpenAI's recent results (including their proof of the quasi-Riemann hypothesis), along with classical methods of Erdős, Granville, Pomerance, and Spiro. It is formally verified in Lean.

This is the second Erdős problem that we've fully resolved using Fractal (coming soon). We will be releasing results on more problems in the coming weeks.

https://t.co/I9AmJy1c0x

[Natalie Trono @Natalie_Trono]

@Domainlogistcom @khulick @PlasmaAI AI!

[Natalie Trono @Natalie_Trono]

Thanks @khulick for including @PlasmaAI!

https://t.co/UdycUv3OKd

[Natalie Trono @Natalie_Trono]

@_andrewturner @nicholas_diao @ScienceNews @khulick

[Natalie Trono @Natalie_Trono]

@nicholas_diao @_andrewturner @ScienceNews @khulick

[Natalie Trono @Natalie_Trono]

@khulick @PlasmaAI @ScienceNews

[Nick Diao @nicholas_diao]

Science News featured our work at Plasma AI.

My co-founder @_andrewturner spoke about how AI is “making math more accessible.” That’s a big part of what motivates us: helping more people explore difficult problems and build on existing research.

https://t.co/ELPbqGIRS3

[Natalie Trono @Natalie_Trono]

@_andrewturner @nicholas_diao ➕🧮👨‍🔬➗➖

[Andrew Turner @_andrewturner]

@nicholas_diao and I had the chance to speak with Kathryn after the OpenAI results were published. The frontier of human knowledge has never been controlled by humans. It pushes itself!

https://t.co/wDSZx81mFr

[Natalie Trono @Natalie_Trono]

@Jon_Hartley_ Amazing!!

[Natalie Trono @Natalie_Trono]

@virolog better safe than sorry

[Natalie Trono @Natalie_Trono]

We’re ready for anything

[Natalie Trono @Natalie_Trono]

@Domainer86 😂

[Natalie Trono @Natalie_Trono]

@IdaeanDaktyl !!!

[Natalie Trono @Natalie_Trono]

@_andrewturner @PlasmaAI I can't find jacob on x 🧐

[Natalie Trono @Natalie_Trono]

@PlasmaAI 🫢