Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Hadamard Factorization and Zeros of Zeta

In course

MATH-313: Number theory I.b - Analytic number theory

The aim of this course is to present the basic techniques of analytic number theory. ...

Description

This lecture completes the proof of Hadamard Factorization and uses it to derive an expression for the zeta function in terms of its zeros. The slides cover topics such as the order of zeros, Jensen's formula, and the behavior of the zeta function near the zeros. The instructor discusses the significance of the zeros of the zeta function and their relation to the function's properties. The lecture also delves into the concept of non-trivial zeros and their implications in number theory and complex analysis.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (32)

Classified information

Classified information is material that a government body deems to be sensitive information that must be protected. Access is restricted by law or regulation to particular groups of people with the necessary security clearance and need to know, and mishandling of the material can incur criminal penalties. A formal security clearance is required to view or handle classified material. The clearance process requires a satisfactory background investigation.

Classified information in the United States

The United States government classification system is established under Executive Order 13526, the latest in a long series of executive orders on the topic beginning in 1951. Issued by President Barack Obama in 2009, Executive Order 13526 replaced earlier executive orders on the topic and modified the regulations codified to 32 C.F.R. 2001. It lays out the system of classification, declassification, and handling of national security information generated by the U.S.

Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century.

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, ... by This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s,1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882. The Hurwitz zeta function has an integral representation for and (This integral can be viewed as a Mellin transform.

Related lectures (33)

Holomorphic Functions: Taylor Series Expansion

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

Complex Exponential: De Moivre's Formula

Covers De Moivre's formula for finding roots of complex numbers and the concept of complex exponential.

Harmonic Forms: Main Theorem

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

Gamma Function and Stirling's Approximation: Mathematical Methods

Discusses the gamma function, its properties, and Stirling's approximation for large factorials, emphasizing their significance in mathematical methods for physics.

Fundamental Theorem of Algebra