1/1² - 1/2² + 1/4² - 1/5² + 1/7² - 1/8² + ⋯
Together with Calegari, Dimitrov and Tang proved this number is irrational... and they just won a $100,000 prize!
It's the biggest advance in irrationality since Apéry showed
ζ(3) = 1/1³ + 1/2³ + 1/3³ + ⋯
is irrational back in 1978. But I should emphasize that it's just the tip of what Dimitrov, Tang and Calegari actually did.
In this video:
https://www.ias.edu/video/arithmetic-some-dirichlet-l-valuesCalegari explains that this is essentially the second "genuinely new" number called a Dirichlet L-value at an integer point to be proved irrational since the 19th century - the first being Apéry's proof that ζ(3) is irrational. Euler had shown
ζ(2n) = 1/1²ⁿ + 1/2²ⁿ + 1/3²ⁿ + ⋯
is a rational multiple of π²ⁿ, so Lindemann's 1882 proof that π is transcendental showed all the numbers ζ(2n) are irrational... but then things slowed down except for Apéry's result, and now this. Nobody knows how to prove ζ(5) is irrational.
For the actual paper, see:
https://arxiv.org/abs/2408.15403It's a 218-page blast of abstraction, but it also proves these numbers are irrational:
1/4² + 1/7² + 1/10² + 1/13² + ⋯ (going up 3 each time)
1/2² + 1/5² + 1/8² + 1/11² + ⋯ (going up 3 each time)
1/1² + 1/7² + 1/12² + 1/19² + ⋯ (going up 6 each time)
and more!
