The Arithmetic of Multiple Harmonic Sums.

Abstract

This dissertation concerns the arithmetic of a family of rational numbers called multiple harmonic sums. These sums are finite truncations of multiple zeta values. We consider multiple harmonic sums whose truncation point is one less than a prime. We derive families of congruences, involving multiple harmonic sums, for binomial coefficients and for values of the Kubota-Leopoldt p-adic L-function at positive integers. Congruences in our families hold modulo arbitrarily large powers of prime. We also set up a framework for studying congruences among multiple harmonic sums, which is related to a framework used in the study of multiple zeta values.