Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Densities of subsets of prime numbers have been studied for hundreds of years, one famous example being Dirichlet's theorem on the distribution of primes in arithmetic progressions. We give a new formulation of Dirichlet's Theorem, and more generally the Chebotarev Density Theorem, which describes the distribution of primes with fixed Frobenius elements in Galois groups. This formula is given as a convergent infinite sum of terms of the form $\mu(n)/n$, where $\mu(n)$ is the classical Mobius function, and the sum includes only those integers $n\geq2$ whose smallest prime factors have fixed Artin symbol. This theorem generalizes work of Alladi from the 1970s in the case of cyclotomic fields.