Continuous Bernoulli Distribution and p-adic Periodic Zeta Function

Abstract

The p-adic Dirichlet L-function or the Kubota-Leopoldt p-adic L-function is a p-adic version of Dirichlet L-function. It is defined by the Mellin-Mazur transform of the Bernoulli distribution μk on Zp× for an integer k ≥ 1. In this thesis, we extend the Bernoulli distribution with integer parameters to a continuous version i.e., a Bernoulli number with a continuous parameter. Also, we generalize the p-adic Dirichlet zeta functions that is defined only for Dirichlet characters to the p-adic periodic zeta functions that is defined for arbitrary periodic functions with values in Cp.