Lectures in course MATH-313: Number theory I.b - Analytic number theory

Covers multiplicative functions, Dirichlet convolution, and the Mobius function in arithmetic functions.

Covers the Euler-Maclaurin summation formula and the method of convolution for evaluating arithmetic functions.

Covers the application of convolution and Dirichlet hyperbola method in arithmetic functions.

Introduces the Abel summation formula and its application in establishing various equivalent formulations of the Prime Number Theory.

Explores Mertens' theorems on prime estimates and the behavior of the Mobius function in relation to the prime number theorem.

Explores the analytic and algebraic properties of Dirichlet series associated with arithmetic functions.

Introduces the Euler product and Perron's formula in arithmetic functions.

Introduces characters over a finite abelian group and explains the proof of the infinitude of primes in arithmetic progressions.

Covers the proof of non-vanishing of Quadratic Dirichlet L-function and reviews basic properties of Gamma function.

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

Covers the functional equation of the Zeta function and the Hadamard factorization theorem.

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

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

\psi(x)

.

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

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