Introduction to analytic number theory 10: The functional equation for zeta function

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Lecture 10 of my graduate course, Introduction to analytic number theory, at the University of Western Ontario, May-June 2023. Analytic continuation and functional equation of the Riemann zeta function (Riemann's second proof): Mellin transform, theta function and modularity.
00:30 analytic continuation of Gamma function
10:00 poles and residues of Gamma function
37:00 completed zeta function
38:27 theta function, zeta via theta, Mellin transform
47:05 modularity of theta
49:00 analytic continuation and functional equation of zeta functions

Masoud Khalkhali

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I love this video; it's very easy to understand and well-explained. Is there an easy way to demonstrate that the Gamma function cannot be zero?

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