In the first half of this talk we will introduce the notion of a generalized prime counting function taking values in a vector space. We will also introduce analogous generalizations of the Chebyshev function and the LCM sequence. Our main theorem relates these three quantities by computing an exact formula for the constant of proportionality in the smallest term of their asymptotic expansion. The formula may be understood as a reformulation of Chebotarëv's density theorem and admits a representation-theoretic interpretation. It has the added feature however of making clear a connection with Landau's prime ideal theorem which is not evident from the usual formulation. Furthermore, densities of primes which naturally arise in consideration with non-Galois fields can also be easily calculated by means of the formula.

The second half of the talk will consider an application to p-adic analysis. We do this by giving a general inequality relating quantities associated with any sequence of elements of a number field. By applying our result to the sequence of coefficients of a power series, we will obtain Dwork's Theorem 3 from his paper on the rationality of the local zeta function of an algebraic variety. By applying our result to the sequence of finite differences of another sequence, we will recover and significantly strengthen some existing results in the context of p-adic interpolation. Speaker(s): Andrew O'Desky (Univ Michigan)