Factorials, prime numbers, and the Riemann Hypothesis

Today we introduce some of the ideas of analytic number theory, and employ them to help us understand the size of n!. We use that understanding to discover a surprisingly accurate picture of the distribution of the prime numbers, and explore how this fits into the broader context of one of the most important unsolved problems in mathematics, the Riemann Hypothesis.

It is this channel's inaugural video! Welcome everyone, we hope you enjoy what you see here, and let us know how we did in the comments!

[claudiocosta]

Wow, if "Euler" was alive, thats the kind of video he would produce: Masterpieces for sharing the beautifull of math to the world.

[williamrhopkins]

My math degree is from the 70's, The use of computer graphics to visualize mathematics is phenomenal. That the likes if Reimann, Euler and Gauss could do so in their heads even more so. Lovely presentation,

[猫会飞]

When I clicked on this video, I wasn't sure if I would last the 55 minutes. As it turns out, that was one of the quickest 55 minutes in my experience of Youtube. Well done!

[marge]

This is the clearest exposition I've seen on this subject on YouTube. Thanks for your hard work and look flawed to seeing more.

[alechowarth]

An amazing lecture! I am in my 80s and I have had a lifelong interest in the PNT. This has given me a deeper understanding than anything I have read previously and inspired me to pursue the topic further.

[abhiramkidambi]

I rarely comment but when I tell you I've gone years of not understanding what exactly the hypothesis (the complex version) had to do with the Primes - this lecture was a very good step in understanding so thank you very much!

[agrajyadav]

You are an awesome individual prof. I did not expect someone to tell in such detail with such quality, about something as beautiful and "complicated" as the Reimann Hypothesis. Thank you professor! God bless u, even though im an atheist

[amritawasthi]

I wish there was as much like buttons as there are the zeta zeroes. The video is beyond incredible. I completed studying apostols ANT prolly like 2 or 3 years ago. And I just visited here today, feels like a joy to freshen the memories. Thank you.

[pythagorasaurusrex]

Great stuff! You are the first one who derives where the Li(x) function really comes from. This is hardly explained in books.

[diribigal]

This was really great! I've seen pieces of this before, but they either go too quickly or I get lost in the careful details of error estimates/complex analysis/whatever that are important if I wanted to work in the field, but obscure the main ideas. Thanks for making this video!

[kriterer]

This is the best math video I think I have watched, and I have watched hundreds

[JustinSmith-ievt]

Brilliant. Thank you for taking the time to create such an engaging teaching video that should make the Riemann Hypothesis understandable to even an interested high school student.

[warisulimam]

This video has so many good sides to talk about that I'm not even gonna attempt it. Please know this work of yours greatly appreciated and is SUPER helpful to amateurs like myself, and I believe to experienced people as well. Thank you!

[Alex_Deam]

Came here from Cracking the Cryptic, have been trying to learn number theory during the pandemic so couldn't avoid clicking on a channel with the name 'zetamath' lol. Excellent stuff, have a 500th like!

[missoss]

Your videos are criminally under viewed.

[MostlyIC]

Totally awesome! I've been watching RH and GRH videos and this is a gem, it gives some background from a different point of view and helps cement the bigger picture. in addition to "a picture [or graph] is worth a thousand words", I have to say how much I appreciate being told both what we know and what we don't know, what we don't know (or what wasn't covered and left to Grad classes that we weren't told about) was so often overlooked in my Undergrad math classes. Well done!

[ilanbar]

If we had school teachers like you the RH would be solved by now 🌼

[MacHooolahan]

Superb.... You spend time explaining in the right places. Cheers from England!

[nindorox]

Thank you for this, it's really great! I've never seen this much context about the Riemann Hypothesis before presented in such an understandable way.

[malicksoumare]

It's been a long time that i have these kind of feelings while watching maths videos... Thank you