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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[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!

[erbgzuwcv]

Next video will be proof of the RIemann hypothesis. I am so hyped rn

[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.

[littlenarwhal]

Great content as usual zeta, this series deserves more recognition

[omograbi]

Why did u stop continuing your series?

[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.

[jedb]

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

[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

[mcpr]

not only are your explanations concise, but you very carefully take pauses to let us think about what you have said. That tactic is overlooked by a lot of really good educators.

[upwardogdownwardog]

Thank you for the incredible work you are doing!

[federicosedilis]

when will you post the next video about the zeta function?

[demezon]

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.

[lowerbound]

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

[ethanjensen]

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

[jameslawson]

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

[noahali-origamiandmore]

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).

[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.

[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.

[muranki]

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 :)