First, what a fascinating article on a truly exciting prospect!

Second, I hope that at some point, it will be explained further what the differences between the 'original', 'arithmetic', and 'geometric' Langlands programs are. I'm not entirely sure which objects are which here, and I think such an explanation will be illuminating for the reader (especially if the Langlands dualities or programs continue to be fruitful and multiply into yet more regions of math).

Third, I wish the authors and reviewers the very best of luck reviewing 451 pages of manuscript, which just on the face of it sounds like a daunting task.

Fourth, I hope to hear further updates on this as mathematicians chew this news and this work over.

I ran the following in my Julia REPL

julia> @time BigFloat(π, precision=20000000); 11.208211 seconds (60.53 k allocations: 2.932 GiB, 0.16% gc time)

Which is pretty good considering I ran this in WSL on my laptop and Mathematica in a different comment took 10 seconds. (Plain Windows took ~26 seconds for some reason?)

I’m too late to update my own comment, but I now suspect BigFloat(π, precision=20000000) may allocate a Bigfloat with precision of two million digits and then store the constant π which has much lower precision in it.

What value does that print?

It definitely does calculate it otherwise it would diverge significantly at lower digits from the Gauss Legendre and Chudnovsky algorithm that I compared it to. It just defers calculation to MPFR, the C library Julia binds to for BigFloat calculations.

Also one additional comment is that you are setting precision in number of bits, not in decimal digits. It obviously runs much faster when it is only computing log2(20000000) not 20000000 digits of precision.

> (Plain Windows took ~26 seconds for some reason?)

Maybe the allocations were the reason?