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Thomas
@aviationdoctor.eth
There's been a remarkable breakthrough toward proving the Riemann Hypothesis, although it still falls well short of fully resolving the conjecture. Larry Guth of MIT & James Maynard of Oxford University (Fields medalist and frequent guest on YouTube's Numberphile channel) have co-authored a pre-print (https://doi.org/m7xb) in which they narrow down a previous estimate (known as the Ingham bound, dating back from 1940!) of the maximum number of non-trivial zeros of the zeta function that could lie in the right side of the critical strip (real part greater than 1/2 but not more than 1). Lay summary: https://www.scientificamerican.com/article/the-riemann-hypothesis-the-biggest-problem-in-mathematics-is-a-step-closer/ Maynard and Guth presenting their results at a recent conference: https://www.ias.edu/video/new-bounds-large-values-dirichlet-polynomials-part-1 and https://www.ias.edu/video/new-bounds-large-values-dirichlet-polynomials-part-2
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Thomas
@aviationdoctor.eth
Any actual zero in that region of the critical strip would invalidate the RH, of course. So mathematicians (who overwhelmingly expect the RH to be true) don't expect to find any. Still, one way to prove the RH is to keep narrowing down this Ingham bound (now the Guth-Maynard bound) for the density of non-trivial zeros in the critical strip until it reaches zero, leaving the critical line as the only real part solution to the zeta function. Terence Tao (who some people believe might be the most likely among all mathematicians alive to possibly prove the RH) acknowledged the breakthrough on Mastodon (https://mathstodon.xyz/@tao/112557248794707738). I now wonder whether a complete formal proof for the RH will come from one of these brilliant minds, or from an automated theorem prover, as this class of tools is improving fast. Maybe I should open a long-burn polymarket for it...
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Sine
@sinusoidalsnail
Hmm maybe it’ll be a combination. For example, automation of small pieces of proof, which mathematicians can then refine and formalize. I think that, similar to how AI is used in other fields, it’ll act as a collaborative partner to mathematicians, as opposed to a full replacement.
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