Lectures related to Functional Equation of Zeta and Hadamard Products

Stirling's Formula and Functional Equation for Zeta

Covers the proof of Stirling's asymptotic formula for the Gamma function and the functional equation of the Zeta function.

Prime Number Theorem

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

Analytic Continuation: Residue Theorem

Covers the concept of analytic continuation and the application of the Residue Theorem to solve for functions.

Unclosed Curves Integrals

Covers the calculation of integrals over unclosed curves, focusing on essential singularities and residue calculation.

Harmonic Forms: Main Theorem

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

Classical Zero Free Region for Zeta and Explicit Formula I

Establishes the classical zero free region for the Zeta function and starts the proof of the explicit formula for

\psi(x)

.

Functional Equation of Zeta

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

Comparison Series and Integrals

Explores the relationship between series and integrals, highlighting convergence criteria and function examples.

Hadamard Factorization and Zeros of Zeta

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

Abel Summation: Analyzing Logarithmic Functions

Explores Abel summation formula, Chebyshev's theorem, and logarithmic functions with practical examples.

Riemann Sums and Definite Integrals

Covers Riemann sums, definite integrals, Taylor series, and exponential of complex numbers.

Distributions and Derivatives

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Matricizations and Alternating Least Squares

Covers matricizations, ALS method, and tensor decomposition, emphasizing the alignment of fibers in matrices.

Proof of Explicit Formula

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

Generalized Integrals: Convergence and Divergence

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Logarithmic Derivative of Zeta

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

Two-loop Computations: External Legs and Internal Propagators

Covers the computation of two-loop diagrams with external legs and internal propagators.

Integration of CnR Class Functions

Explains the integration of Taylor series for CnR class functions.

Integrals of Functions

Covers integrals as the reciprocal function of the derivative and explains displacement integral of velocity and variation of speed and acceleration in rectilinear motion.