The Zeta function, particularly the Riemann Zeta function, is a complex function pivotal in number theory and mathematical analysis, with profound implications for the distribution of prime numbers. It is defined as a series that converges in the complex plane and has a continuation that is analytic except for a simple pole at one.

Zeta Function

Zeta function

The Riemann Zeta Function is a complex function that plays a pivotal role in number theory, particularly in the distribution of prime numbers. Its most famous unsolved problem, the Riemann Hypothesis, conjectures that all non-trivial zeros of the function have a real part of 1/2, which has profound implications for mathematics and cryptography.

Riemann Zeta function

complex function

Number theory is a branch of pure mathematics devoted to the study of the integers and integer-valued functions, exploring properties such as divisibility, prime numbers, and the solutions to equations in integers. It has deep connections with other areas of mathematics and finds applications in cryptography, computer science, and mathematical puzzles.

number theory

Mathematical analysis is a branch of mathematics focused on limits, continuity, and the rigorous study of functions, sequences, and series. It provides the foundational framework for calculus and extends to more complex topics such as measure theory and functional analysis.

mathematical analysis

distribution of prime numbers

series that converges

The complex plane is a two-dimensional plane used to represent complex numbers, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This visualization allows for a geometric interpretation of complex number operations, such as addition, multiplication, and finding magnitudes and angles.

complex plane

Analytic continuation is a technique in complex analysis that extends the domain of a given analytic function beyond its initial region of definition. This process allows for the exploration of the function's properties and behaviors in a larger domain, often revealing connections between seemingly unrelated functions or domains.

Relevant Degrees

Log in to see lessons