Lecture

Logarithmic Derivative of Zeta

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Completes the proof of Hadamard Factorization and uses it to derive an expression for the zeta function in terms of its zeros.

Covers the definition and properties of the Riemann Zeta function, including convergence and singularities.

Covers the functional equation of zeta function and Jensen's formula in complex analysis.

Explores uniform convergence of series of functions and its significance in complex analysis.

Explores the proof of the Prime Number Theorem and its implications in number theory.

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Covers the proof of the explicit formula for the non-vanishing of the zeta function at the 1-line.

Explores the Riemann zeta function, its properties, applications, and analogies in number theory and algebraic geometry.

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Explores the logarithmic derivative of the Zeta function using the Hadamard factorization.

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Covers the Hadamard factorisation theorem for entire functions of order at most 1.

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Explores the Euler product theorem and Perron's formula in number theory and complex analysis.

Explores the convergence of a series parameterized by alpha and its conditions for convergence.

Covers the application of the residue theorem in calculating integrals on closed curves in complex analysis.

Covers the concept of Region of Convergence for signals with single poles.

Covers the proof of Von Mangoldt's formula and the Prime Number Theorem using Zeta functions and pole analysis.

Covers the convergence criteria for sequences, including operations on limits and sequences defined by recurrence.

Covers the correction of a mock exam for Analysis I, focusing on sequences, series, and limits.