Dirichlet L-function

In mathematics, a Dirichlet L-series is a special kind of math function. It looks like a sum that uses something called a Dirichlet character and a complex number. These functions are very useful in number theory, which is the study of numbers and their patterns.

These functions are named after Peter Gustav Lejeune Dirichlet, who introduced them in 1837. He used them to prove an important theorem about primes in arithmetic progressions. Dirichlet showed that these functions never become zero at a special point, which helped him prove his theorem.

By using a process called analytic continuation, mathematicians can extend these functions to work across the whole complex plane. Depending on the Dirichlet character used, these functions might have a simple pole at a certain point or they might be entire, meaning they have no poles at all.

Euler product

A Dirichlet character is a special kind of rule that works with numbers. Because of this rule, the L-function can also be shown 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 these special math functions called L-functions, things become easier if we assume the character used is "primitive." This means we can understand more complex cases by using a simpler one.

For a special type of character called the principal character, the L-function can be linked to something called 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 across the whole complex plane. It connects the values of L(s, χ) to the values of L(1−s, χ̄).

When χ is a special type of character called a primitive character modulo q (where q is greater than 1), the functional equation can be written in a specific form. This form involves several mathematical symbols and functions, including the gamma function and a special value called W(χ). The equation shows how L(s, χ) relates to L(1−s, χ̄) through these symbols.

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 entire functions, meaning they are smooth and well-behaved everywhere in the complex plane. If q equals 1, then L(s, χ) becomes the Riemann zeta function ζ(s), which has a special point at s = 1.

Main article: functional equations of L-functions

Zeros

In math, there are special numbers called zeros where a certain kind of math 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 certain 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 expressed 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 can be built from the Hurwitz zeta function. This shows that the Hurwitz zeta function and Dirichlet L-functions share important mathematical properties.

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