I remember as a teen when my parents got me a pc (they could Ill afford) as an upgrade from the, by then 10 years old, hand-me-down ZX Spectrum.
I was in awe at how many primes I could find on my cyrix 333Mhz on my own noddy code
Left it running all day when at school.
Never found any thing of value but I learned how to Beowulf cluster and learned a lot about arbitrary integer implementation
I suspect somewhere in the Riemann-Zeta function is a reciprocal that only cancels out all the imaginary terms once and only once, thus any number with more than one factor wouldn't cancel out.
I haven't the math skills to find that reciprocal.Reply
This fascination with primes is has, of late, seemed to be a little misplaced to me. Primes emerge as a property of the number system that you have.
The number system itself is an an approximation of the natural world where actions like addition, subtraction, multiplication and division exist. The natural world in question is, as far as we know today, what we humans observe.
Now, imagine a second world which was excellent at approximations to the extent that their entire Standard model depended on differentiation and integrations. Would they care as much about primes and try to fit everything into “beautiful” math? I don’t know the answer to this but I’ve found this thought experiment useful to put our maths in its place.Reply
Is there a reason we're obsessed with primes beyond aesthetics? Why does this set of numbers garner all the headlines as opposed to some other arbitrary integer sequence like the Recamán numbers  ?
If tomorrow someone discovered a closed-form equation for the nth prime, how would mathematics/the world change?Reply
I love how every quote by the mathematician clearly states that this does not bring us appreciably closer to the Riemann hypothesis, or that it is very unlikely, while the bulk of the article seems to imply otherwise.Reply
What impact would this have on elliptic curve cryptography?Reply
Starting to think quantamagazine writes headlines this vague as a conscious clickbait strategy by now because these pull the biggest audience.Reply
Two inaccuracies in the article. For purposes of simplicity, the author writes "But the Lindelöf hypothesis says that as the inputs get larger, the size of the output is actually always bounded at 1% as many digits as the input." But this is a case where simplification goes too far. What Lindelof says is that the size of the output is always bounded by ε% as many digits as the input, for ANY (arbitrarily miniscule) ε > 0.
Second, the subtitle "Paul Nelson has solved the subconvexity problem..." is strange. The subconvexity problem, for a given L-function, is to lower the percentage described above from 25% to "any positive number"; in other words, to bridge the gap between the convexity bound (which is "trivial") and Lindelof (a consequence of GRH). The only way that the statement "Paul Nelson has solved THE subconvexity problem..." could maybe be accurate is if Nelson proved the Lindelof hypothesis for "all" L-functions. Which is far from the case. (What makes subconvexity so interesting, as Nelson says in the article, is that it is a problem where you can make partial progress towards a particularly important consequence of GRH. And Nelson's result is exactly that: partial progress.) More accurate would be "Paul Nelson has made significant progress on the general subconvexity problem."Reply
It might be that the Riemann Hypothesis will be the first of Math's "Hard Problems" that will be solved by AI before humans...Reply
Young enough for a Fields? There is a ceremony later this year, but it's probably too soon. So he could be awarded a Fields in 2026 if he's not yet 40. He received his PhD in 2011.
Also, according to MathGenealogy, his grand-grand-grand-grand-grand-grand-grand-grand thesis advisor was none other than Poisson himself. Poisson's advisers were Lagrange and Laplace. Lagrange's was Euler. Euler's was Bernoulli. Bernoulli's was another Bernoulli.
You can go back to the late 15th century to find mathematicians with "unknown" advisors.Reply