Dedekind eta function
Introduction
The Dedekind eta function, named after the German mathematician Richard Dedekind, is a key concept in the field of number theory and modular forms. It is a modular form of weight 1/2 and is defined on the upper half-plane of complex numbers. The Dedekind eta function plays a significant role in various mathematical theorems and equations, including the Rademacher's formula for the partition function and the Jacobi triple product identity.
Definition
The Dedekind eta function is defined for all τ in the upper half-plane, which is the set of all complex numbers with positive imaginary part. The function is given by the infinite product:
η(τ) = e^(πiτ/12) ∏_(n=1)^∞ (1 - e^(2πinτ))
This product converges for all τ in the upper half-plane.
Properties
The Dedekind eta function has several important properties. It is a holomorphic function on the upper half-plane and satisfies the transformation law:
η(-1/τ) = √(-iτ) η(τ)
This property is a reflection of the modularity of the eta function. Another key property of the Dedekind eta function is its relation to the Jacobi theta function and the modular discriminant.
Applications
The Dedekind eta function is used in various areas of mathematics. In number theory, it is used in the proof of the Rademacher's formula for the partition function. In the theory of modular forms, the Dedekind eta function is used to construct modular forms of higher weights. The eta function also appears in the study of elliptic curves and string theory in physics.