# Dirichlet L-Functions and Their Zeros

Dirichlet L-Functions and Their Zeros

Dirichlet -functions are important in number theory. For example, -functions are used to prove Dirichlet's theorem, which states that the arithmetic progression () contains infinitely many primes, provided and are relatively prime. The zeros of -functions can even be used to count how many primes less than there are in arithmetic progressions.

L

L

qn+a

n=0,1,2,…

q

a

L

x

This Demonstration graphs Dirichlet -functions along the line in the complex plane (the so-called "critical line"), and highlights the zeros that are encountered. Zeros occur where the real part (blue graph) and the imaginary part (red graph) are simultaneously 0.

L

1/2+it