An Introduction to Artin L-Functions

An Artin L -function is a generalization of the Riemann zeta function and the classical Dirichlet L -functions. Just as the Dirichlet L -functions are useful in the study of primes in arithmetic progressions, so are the Artin L -functions useful in the study of the decomposition of primes in algebraic number fields. In contrast to the classical objects, we still do not have analytic continuation of these objects in the general setting. If we did, this would have profound consequences in the study of prime number theory, especially to various forms of the effective Chebotarev density theorem, which can be viewed as the most general form of the prime number theorem.