'Sensational' Proof Delivers New Insights into Prime Numbers(quantamagazine.org)
quantamagazine.org
'Sensational' Proof Delivers New Insights into Prime Numbers
https://www.quantamagazine.org/sensational-proof-delivers-new-insights-into-prime-numbers-20240715/
4 comments
I don't think this is anything special about primes or e - if you replace prime(n+1) with just (n+1) itself you get the same sort of patterns. But it is something to do with approximations of irrationals by rationals - you might want to look into continued fractions. Try replacing e with a rational number a/b (say 8/3 or 11/4); then you get b horizontal-ish lines, corresponding to the different remainders of n when divided by b. So the pattern you get with pi isn't "as good" because pi is famously close to 22/7.
In that case, you might get the best results with Liouville numbers.
This is really well explained. The video helped me to finally understand the connection between zeta function and prime distribution.
Dup: 'Sensational' Proof Delivers New Insights into Prime Numbers
https://news.ycombinator.com/item?id=40968164
I'm curious what the specific bounds were before and after this proof.
Perhaps this will help? https://mathstodon.xyz/@tao/112557249982780815
The primes are somewhat evenly spaced with this transformation, I'm the author of it.
A342730: a(n) = floor((frac(e * n) + 1) * prime(n+1)).
https://oeis.org/A342730/a342730.png
Instead of e, I have tried other constants such as pi, but it doesn't look as good. I guess there is another constant that makes the distribution look even nicer.