Complex analysis: Zeta function functional equation

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This lecture is part of an online undergraduate course on complex analysis.

We use the residue calculus to prove the functional equation of the Riemann zeta function (following Riemann's first proof).

(Warning: I got really confused trying to sort out the signs in this video, so do not rely on them.)

Solution to exercise (in rot13): cv bire fva cv f

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One of the best thing about this channel is knowing one of the greatest mathematicians also get confused with signs and some definition often not even trying to hide it. Best respect.

gunhasirac

"If I don't tell you which way I'm going around the contour you can't catch my sign errors" lol

abhinavnatarajan

This really is the best math channel out there now. We are so lucky.
(And Borcherds cannot but be pretty hilarious at times. Idkw, but accidental comic relief is so much more satisifying than all the people out there writing scripts.)

indiscernables

You have quickly risen to my favorite math channel. Top notch content bruh!!!

brian

Thank you for your explanations of the contours. I got totally lost reading Edward's "Riemann's zeta function" for this exact reason.

theflaggeddragon

I love how it's not at all explained rigorously, but still exact

ethanjahan

Profwssor, if I am not mistaking, this is not a Bromwich contour (which is used in inverse Laplace) but a Hankel contour.

ricardoa

How do you reconcile the fact that for the integral over the small circle to vanish Re(s)>0 and later for the integral over the big arc to vanish Re(s)<0?

ricardoa

This fascinated me years ago, when i was able to relate the Sine function in terms of reflections of Gamma and Zeta. Creepy almost... that in a way, prime numbers are a subtle hidden element in the Trig functions.

kharnakcrux

I feel confused, I keep getting sign errors or are there sign errors in the video?

Salmanul_

At 6:55 should the fraction be equal to the geometric series 1 + e^(-z) + e^(-2z) + ... ? Did you forget the 1 or does it not matter

uy-gedm

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