Prime number races

2 is the oddest prime? It's the only one that isn't odd :D

Rion Boom Crabhands Keon

I feel like the Number Theory graphics person missed something when they didn't present each number as inset in a circle. 371119 run together in a confusing mess, whereas ③⑦⑪⑬ are relatively legible.

Another fact about primes that may be related to the Chebyshev bias is the Littlewood bias (I don't know if it's called that, by the way!), namely the bias of the prime counting function π(x) being usually smaller that li(x). In fact, I understand that π(x) exceeds li(x) infinitely many times, much like π₁(x) exceeding π₃(x) infinitely often. Are these facts related, and is there a known fraction of the time in which π(x) exceeds li(x)?

David Terr

Fascinating video! I've known for some time about this bias, which is called the Chebyshev bias, and in fact holds for other modular classes of primes as well, such as primes congruent to 1 and 2 modulo 3, the latter of which has an edge. But ever since I first learned about this bias, I've been mystified as to why it holds. You explain it quite well in this video! I wonder how one my go about proving this fact, or other related ones.

David Terr

What are the points where it drops to zero? Some number in this region must be special in some sense. To be fair this is just an instant reaction I will go an look at the referenced paper. Thanks as always for an nteresting video.

Fascinating. This seems worth turning into a more canonical 3B1B video with all the nice graphics working for you. Some of which are presented here, but not coupled cleanly to your voice over.

Mike Jarvis

Yes, precisely, it's taking the logarithm of an Euler product.

3blue1brown

Oh nice! That makes sense, thank you :)

Aman Karunakaran

I think, you expand the series using the Euler product and if the condition that was mentioned in the video holds, then the factors resemble a geometric series, and so you can replace them by the corresponding fractions. The logarithm of that product is then equal to the sum of the logarithms of the individual factors and if you replace those with the Mercator series, you should be done.

Richard Kalhöfer

Does the "number theorist's favorite way to take a logarithm" have an official name? I tried looking it up in various ways to no avail, and I skimmed the paper you reference to see if anything was there but I didn't see anything that looked similar (although I could have missed it, the paper was long and I skimmed kind of quickly!) I'm very curious about the proof sketch for such a theorem, it seems so out of left field.

Aman Karunakaran