Dirichlet L-function

In mathematics, a Dirichlet L-series is a special kind of math function. It is a sum that uses something called a Dirichlet character and a complex number. These functions help us study numbers and their patterns.

These functions are named after Peter Gustav Lejeune Dirichlet. He introduced them in 1837 to help prove a big idea about prime numbers.

Mathematicians can use a method called analytic continuation to make these functions work everywhere in the complex plane. Depending on the Dirichlet character, these functions might have a simple pole at one point or they might have no poles at all.

Euler product

A Dirichlet character is a special rule that works with numbers. Because of this rule, the L-function can be written in another way using something called an Euler product. This works when we look at numbers where the real part of s is greater than 1. The product uses all prime numbers.

Primitive characters

When studying special math functions called L-functions, things are easier if we assume the character used is "primitive." This means we can understand harder cases by using a simpler one.

For a special type of character called the principal character, the L-function can be linked to the Riemann zeta function. This shows how these math ideas are connected.

Functional equation

Dirichlet L-functions follow a special rule called a functional equation. This rule helps us understand the values of these functions everywhere.

When χ is a special type of character called a primitive character modulo q (where q is greater than 1), the functional equation has a specific form. This form uses several mathematical symbols and functions, including the gamma function and a special value called W(χ).

The functional equation also introduces another function, Λ(s, χ), which is closely related to L(s, χ). For this special case, Λ(s, χ) and L(s, χ) are smooth and well-behaved everywhere.

If q equals 1, then L(s, χ) becomes the Riemann zeta function ζ(s).

Main article: functional equations of L-functions

Zeros

In math, there are special numbers called zeros where a pattern equals zero. For the Dirichlet L-function, there are no zeros when a certain part of the number is greater than 1.

When this part is less than 0, there are zeros at specific negative whole numbers. If a rule equals 1, the zeros appear at -2, -4, -6, and so on. If the rule equals -1, the zeros appear at -1, -3, -5, and so on.

The rest of the zeros are found between 0 and 1 on the real part of the number. These are called non-trivial zeros. They balance around the middle line at 0.5. A big guess in math is that all these non-trivial zeros lie exactly on this middle line.

Relation to the Hurwitz zeta function

Dirichlet L-functions can be written using a special math function called the Hurwitz zeta function. For a whole number k that is 1 or larger, Dirichlet L-functions for characters related to k come from the Hurwitz zeta function. This shows that these two functions have important math properties in common.

The Dirichlet L-function for a character linked to k can be expressed in a special way using the Hurwitz zeta function. This connection helps mathematicians understand these functions better.