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This lecture explains the process of approximating the zeta function using a smooth compact function. It covers the transformation of vectors, spectral decomposition, and the importance of compactness in the analysis. The instructor demonstrates the application of the functional equation for the integrity of data, emphasizing the role of convexity. The lecture delves into the details of the approximation method, showcasing the integration over compact sets and the significance of the Fourier base in understanding the proof. The discussion also touches on the concept of feedback and the principles involved in the proof, providing a comprehensive roadmap for interpreting mathematical documents.