Complex Integration and Finding Zeros of the Zeta Function

In this video we examine the other half of complex calculus: integration. We explain how the idea of a complex line integral arises naturally from real definite integrals via Riemann sums, and we examine some of the properties of this new sort of integral. In particular, we consider some complications that arise when trying to apply the fundamental theorem of calculus to complex functions.
We then bring these ideas to the central focus of this series: the zeta function and the Riemann hypothesis. By the end of the video, we will be able to use complex integrals to approximate the location of the zeroes of the zeta function (or those of any other complex function for that matter)!
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Also, check out our second channel @zetamathdoespuzzles , where we stream sudoku and other puzzle content three times a week.
Animations in this video were produced using Manim Community Edition. For more information, go to manim.community

[@sherifffruitfly]

People always explain the real case as "there's only 2 paths from a ->b" - this seems wrong to me. There's bunches of paths - imagine a drunk walking home from the bar to his house. Even constrained to a line, he could walk forward 2 steps, then back 1 step, then forward 3 more, etc. - until he gets home. Lots of paths, constrained to the [a,b] interval. It's of course easy to prove that you're adding & subtracting integral-pieces, so the total sum remains the same. But that doesn't mean there aren't lots of paths - there are, it's just that they result in the same integral.

[@AlessioDellaMotta]

These are some of the best math videos I have watched in a while, especially about complex analysis! Looking forward to the next ones!

[@liamturman]

Hey man! These are some of the highest quality math videos I have ever seen. Amazing work, I’m so excited for the next video, whenever that is.

[@littlenarwhal3914]

Great content as usual zeta, this series deserves more recognition

[@jameslawson9754]

This is one of the greatest maths videos I have ever seen

[@upwardogdownwardog]

Thank you for the incredible work you are doing!

[@zetamath]

Wow, thank you so much! It means a lot!

[@flmbray]

I just finished watching all of your videos made so far... I was a AP Calculus BC student in high school but chose a different direction at university (electrical engineering), so even though I really love math, I never got into these advanced topics relating to zeta function, and I've never really quite understood it in the various videos I've seen. But watching your videos has made it so easy to understand. I really love your long-form content, giving time for the ideas to really come together. It's been 5 months since this video, and about the same since the video before that, so I'm really hoping the next video is close!

[@dancingacidpanda]

Love your content! This is by far the best introduction to complex analysis I have ever seen. I feel like I am finally getting it :D

[@zetamath]

Thank you so much!

[@Savahax]

Strongly agree mate

[@lowerbound4803]

Wholeheartedly thank you for making all these videos. You deserve my highest respect! 😻

[@demezon6572]

Thanks a ton. When I worked on my thesis, a wee 40 y's ago, I could have used your explanations-to get to the results all the quicker. Now I know the details. You're a professor par excellence.

[@davidblauyoutube]

7:40 When I was learning complex analysis, I was taught about branch cuts and Riemann surfaces. If you "cut" the complex plane along the negative real axis, "lift" the upper half and "lower" the bottom half, you get a helicoid. You can "glue" this to another, similar helicoid (on which the log takes values incremented by 2πi) seamlessly so that analytic continuation travels along the path in the video and ends up at a different location (on the glued surface) than you started, with a different value. By gluing together many helicoids, you can account for the multiple values taken by the log function in a completely consistent way.

[@jedb872]

Fantastic video! You make a difficult subject very accessible. (And thanks for the call out, that was way cool)

[@zetamath]

Thanks so much for giving us the support before we even asked for it. It was much appreciated!

[@colonelmoustache]

It is a pain for me not to donate, but gosh these videos are wonderful. I've never seen a video so well animated anywhere. The rhythm is perfect, the explanations are clear and are not just dumbed down examples. Some quick proofs to really convince are shown The amount of time and effort put into this clearly pays up for the waiting time Man this is perfect, I just killed my evening watching all at once

[@__hannibaalbarca__]

This is only beginning; but very fundamental to go deep in complex analysis, ...

[@solokototelluride]

You have the ability to explain complex topics in simple ways. I see that you do not often release new content, but I do look forward to further releases.

[@Ivan_1791]

This video was wonderful, I hope you post more content in the future.

[@TheLethalDomain]

Who would have thought that one method of studying the Riemann hypothesis would lead to the best introduction to Complex Analysis I have seen on KZbin. I can't wait to see how well you execute Cauchy's Residue Theorem.

[@you8sand]

I remember my professor brushing over some of this material in my electrical engineering undergrad specifically for inverse Laplace Transforms, but the way you work through and present it is extremely clear and helps fills some gaps in my understanding. Thank you for this content; love it!

[@ethanjensen7967]

I'm excited for the next episode!! I'm starting graduate school this fall and I finally get to take complex analysis. I've spent about a year learning math in the field of Modular Forms and Elliptic functions, and a couple of the big proofs require complex analysis that I haven't learned. This series helped me organize what I know before I begin my next class

[@zetamath]

It's great to hear a grad student is finding this material useful. That was one audience I had in mind when laying this out. I'm hoping that these videos provide good scaffolding on which you can fill in more rigorous definitions!

[@michaeldevilangel]

@@zetamath hi there, if it makes you happy, I study towards MSc in Mathematics and still watch your videos:) they are really great! Best for revision and for visualizing certain concepts. Can you do some stuff on spaces? You know for approximation theory. Vector space, Banach space, Hilbert space.... Anyway, thank you very much for your work.

[@rogerkearns8094]

@@zetamath I'm still looking forward to the next episode, too.

[@peachesaupear8455]

I was just rewatching your videos on analytic continuation and the Basel problem earlier this week! I somehow knew the series was about to continue!

[@bobasawrus]

Haha - analytic continuation

[@mattkerle81]

Yay! I just found your channel and watched your earlier Zeta videos two weeks ago, so happy to see this new one out! I'm really hoping you end up covering the sum rho of the zeros of the Zeta function ends up in the formula for the error correction of the prime counting function! Please keep the videos coming!

[@WhattheHectogon]

I just binged all of your videos in the last couple of days, and you do an excellent job. I'm enthralled by your explanatory style, and I can't wait for the next addition to this series!

[@EthanBradley1231]

This is astounding, both the math and your presentation of it. I've seen plenty of your puzzles over on Cracking the Cryptic, so I knew I was in for something good when I found your channel, but this is far better than what I expected. Thank you for making this.

[@mehdibenmassaoud4681]

Amazing video with exceptional explanations. I'm sure everyone would appreciate to have the next video quicker (6 months between each 2 episodes is too long) ! Can't wait !

[@physira7551]

Ive learned a whole semester worth of complex analysis in just one video! I dont know how to thank you. Please keep making these kind of videos

[@francescocatalanotti7506]

You've made only few videos. But every single one of them should deserve an inifinite amount of likes and comments.

[@taj-ulislam6902]

Amazing video. Very clear and presented like a true professional. A complex subject tackled well.

[@bobingstern4448]

This is magical timing! I finished watching your analytic continuation video yesterday and you post this one today!!!!!

[@-_-_-_-_]

I discovered your channel a few days ago and I just wanted to say that your channel is the single best math channel I've ever seen. 3B1B, Mathologer, Numberphile and all the other math channels can't hold a candle to the perfect ratio of rigor and accessibility your channel has. I can already say that with absolute confidence with only five videos uploaded. I have never learned all of these things that you're talking about since I have just graduated from high school and I can assure you that your videos are perfectly accessible to that level of understanding of math.

[@SeeTv.]

Amazing video as always! It's really refreshing to see high level math in such an easy to digest format. Can't wait for the next one.

[@ashutoshyadav1270]

When I'll some money after I start working I will definitely donate to this channel.

[@achilles5716]

Your videos always get me very fascinated about maths. I think you do a phenomenal job at explaining the logical processes leading to solving the problems. I'm very excited to do maths at university and this video is just another reminder as to why.

[@muranki4150]

wow! the editing and explanations in this video are amazing! I'm a highschool student and i actually feel like i learned something. Thank you so much for making this so accessible :)

[@elfumaonthetube]

I've been looking for this kind of details for a long time. To me, this video and the others in the channels are the best on this topic found so far on KZbin. Looking forward to watch the next one!

[@sylverfyre]

Really appreciated this series! As someone who didn't get much formal education beyond the first few college level calc courses, but enjoys math, it's always been a gap in my knowledge of "the zeta function is tied to the primes" and other things about the reinmann hypothesis, without really understanding how one actually executes the analytic continuation to make those conclusions.

[@veechimo]

Of all the math channels I've seen, I like your style of explanations (and animations) and logic/reasoning the best. Really looking forward to the next video in this series!

[@tinic1]

I'm commenting to help the algorithm. This is an amazing series. I only have an amateur understanding of mathematics and I'm learning a ton. The visuals are incredible and the pacing is perfect to allow for new concepts to sink in. Keep them coming!

[@Hazzanger]

Appreciate the hard work you're putting out! A super fascinating function

[@anjandwaraknath9530]

Great content, thanks for taking the time to make these videos. Looking forward to the next one!

[@zacbrennan807]

Just found this series and have been really enjoying these high quality dives into something totally outside my research area. As one piece of advice, I would love to see these videos with closed captions. It really makes a difference to enable that accessibility feature on these high quality, educational videos. Seriously though, great work.

[@adsoyad2607]

Can't tell you how happy understanding this stuff makes me. I'm really looking forward to watch the next video!

[@b.c.johnson1191]

I'm counting on the follow-up to this video. It was very well done. I hope my $ helps motivate you. I am going to watch the previous ones.

[@Rugjoint]

worth the wait but i already want the next episode, really great content!

[@chrissch.9254]

I‘m an analytic number theorist by myself, and I really love your videos - extremely well explained with a logical thread running through the complete series. I‘m already very much looking forward to the next video - keep on your fantastic work!

[@fernandonc3000]

Excellent video as always, fortunately KZbin has Channels like that, where you can actually learn something interesting and not trivial in less than oe hour! Keep doing this amazing job!

[@jaeimp]

These lectures are phenomenal! Thank you! You got it right by having the right tempo in your exposition, doing away with the annoying music in other great YT channels (e.g. 3B1B), and showing clean and uncluttered beautiful LaTeX expressions blending into each other at the precise right moment. So good!

[@MrCreeper20k]

I majored in math and computer science separately in undergrad and i'm going into a software engineering job, but I still have such a strong passion for math! Seeing videos like this and some of the independent studying I have done make we want to do math research! I'm trying to figure out the feasibility of doing research outside of academia lol.

[@wawangsf]

Speaking from someone who have walked those path (15 years in computer science) I would encourage you to follow your passion. Despite of the fact almost all contributors to modern physics have a PHD degree (even Einstein) I do believe it's feasible to make discoveries outside academia. Also Im convinced dedication surpasses all natural gifts.

[@satioOeinas]

@@wawangsf Can you elaborate? Would appriciate it.

[@wawangsf]

@@satioOeinas If you find a field that you are passionate about you ought to devote your time and energy to it. I don't believe you need to be very gifted nor obtain a PHD degree through Academia to be successful in that field.

[@LANless]

Good stuff! I took my only course on complex analysis ~10 years ago, and this series has been a really pleasant and entertaining way of refreshing my memory.

[@LANless]

Actually, this bit about verifying the Riemann Hypothesis numerically (how many zeros before you say 0.500000* =1/2) reminds me a lot of the ongoing arguments about the cosmological tensor-to-scalar ratio - how small does the upper limit on an observable have to be before you argue it is identically zero? Funny that we keep encountering the same questions in very different areas. Thank you for putting this up!

[@libercode9886]

This is amazing! Not only gave me grasp at understanding the Zeta function and complex analysis, it also broaden my understanding of line integrals and made me understand them much better than my engineering degree ever did. Thanks a lot! I'm very excited for the next episode!

[@topquark22]

I studied calculus on multidimensional manifolds. I will just say, It gets much more fun.

[@ericturnbull1500]

Fantastic video. I just finished a degree in computer science, and this series is really inspiring me to go back and learn all of the math that I've missed. I never thought I'd be this excited to go and buy a complex analysis textbook

[@omograbi]

Why did u stop continuing your series?

[@GordPollock]

These videos look like a lot of work. It's not a mystery to me. I'm still hoping he gets back to releasing them.

[@NeTfLeXr6s]

I've recently written my bachelor's thesis on a subject which at it's core is complex analysis. This is one of the best intuitive introductions to the topic I've ever seen. Of course, you are missing a lot of (technical) details, but as far as a first introduction is concerned, this is gold.

[@zetamath]

You're totally right about omitting (loads of) technical details, but my overall goal in these videos is twofold: 1) for one audience, this is all they want to know, and it lets them understand something deep and difficult with a modest time investment. 2) For those who are learning this stuff, to provide a scaffolding to attach the details to as they learn them. They can learn the overall structure of the subject, and then fill in the details as they go along. I also see a lot of people (like you) who learned about this stuff in the past in one way or another and enjoys these videos as a way to reconnect with it and maybe put it in a new context. That's cool too!

[@angeluomo]

Excellent video. Clear and illuminating. Many thanks for all the hard work in putting this together.

[@markstavros7505]

These videos have been amazing! I can't wait for the next one! Subscribed and liked! God bless! 😊👍

[@jogloran]

I previously thought Mathemaniac’s explanation of complex integration was the most cogent on KZbin, but now I’ve seen yours, I have to say this takes the prize, hands down.

[@bjnobre]

Great channel!!! Just waiting for the next video on this series.

[@alijoueizadeh2896]

Thank you for the best presentation of this issue I have ever watched.

[@ANTONIOMARTINEZ-zz4sp]

Looking forward to watching the next video! The best explanation I've ever received about this topic! Thank you!

[@snake4eva]

@zetamath Love your explanations. I have no idea why your videos don't have millions of views. I recently started to show interest in the Zeta function for the first time and I have background with complex analysis. However, your videos make it much easier and gives a new perspective on how to systematically derive complex valued functions in an intuitive way. I hope you continue your videos and resurface because your work is much needed.

[@noahali-origamiandmore2050]

Hey, zetamath. I've really enjoyed watching your videos so far. When is the next episode coming out? I'm curious to know what will happen next. I would love to see how we solve this roadblock against finding the zeros of ζ(s).

[@rewixx69420]

im wating too

[@bowtangey6830]

Please tell me this isn't the last!

[@JustPassingBy_]

Your videos are amazing, with such clear explanations, and love the fact that you sometimes add exercises to the viewer to let us check how much have actually learned, also I don't know if you could recommend me some books to continue learning about analytic number theory, since you showed me how interesting it can be.

[@benhsu42]

Thanks you! "Analytic continuation" has been on my to learn list for a long time and you explained it perfectly

[@Deutschebahn]

This is so great! Really keen for next video, curious about when it might be coming out?

[@ruscul7155]

super underrated channel.. thanks alot for these videos!

[@mcarston]

Outstanding video, I hope you make more in this series soon

[@franciscoabusleme9085]

Excellent quality, it has been a while since a bingwatched hours of math videos. Thank you for motivating me to pick a complex analysis book again. Please tell me you are releasing another video!

[@1224chrisng]

This really helped me understand EMF in Physics. Thank you

[@bobasawrus]

Your inspiration may get me back into that complex analysis class - AGAIN!

[@jakobj55]

Thank you so much for this incredible content. I´m looking forward to the next episode.

[@yazeed0ps3]

Loved the series! Hope to see the next one soon 👀

[@chrissedwick7748]

Oh and please keep making these. I like how you visualize your questions and responds. You also have a nice speaking voice as well.

[@BassamShow]

We are waiting for the next video. please, do not be late It's been six months now And thank you for this great work

[@pion137]

Fantastic explanations! Can't wait for the next video!

[@thomasblok2120]

Very clearly explained. Can't wait for the next video!

[@ibite100]

greatly underrated channel. all your videos are high quality and well explained. keep up the great work.

[@rogerkearns8094]

I've wondered about a lot of this for a long time, thank you.

[@malicksoumare370]

Definitely my channel of the year. Great work!!!

[@marchenwald4666]

Your videos are GOLD ❤

[@Raspberry_aim]

Phenomenal content- thank you for making these videos!

[@petergregory7199]

Your videos have perspicacious moments that warm the cockles of my heart.

[@jakubantoniak732]

I'd love to see next episode :) Great video!

[@josephlorizzo8997]

i love those videos so much, i still have to do vector calculus but the way you are explaining this is making me understand anyways, i love complex analysis and complex numbers in general

[@mMaximus56789]

Can't wait for the next video, this is amazing!

[@thatotherdavidguy]

eagerly anticipating the next one!

[@LittleGnawer]

Great presentation of a "complex" topic . I like the high level animations in this video, too.

[@vladthemagnificent9052]

It's about time to drop the next video. The explanations are on top!

[@AdhyyanSekhsaria]

Thanks! Looking forward to the next video!

[@BiplobHossainSorker]

waiting for next one!

[@joaquincarrasco3419]

Great video. Hope to see more in the future.

[@brighamhellewell6479]

This is now my favorite math channel

[@MacHooolahan]

Great stuff as usual - so well explained :)

[@TheRmbomo]

Thank you for making the video. Enjoyed your explanations ^^

[@VikiVat]

can't wait for the next video! nice job!

[@AllemandInstable]

you brought back many memories long forgotten about analytic functions, havn't encountered those for years and therefore could not remember many things

[@roygalaasen]

I am assuming that the next video is lost in the void then? 😂 It may seem like I am laughing, but have a closer look at those tears. But seriously, your video series is so good that it sparked an interest in the topic, still hoping there will be a follow up with poles as you promised. (Not holding you accountable for any promises, just grateful for the massive job you already have done.)

[@TheJara123]

Brilliant man, please post more often videos!! I think YT AI wants that too!!

[@inflivia]

Very nice series. More please!

[@mathsforiit-jee9730]

I hope this video would have come when i was studying in my college

[@shameer339]

Great explanation 👏