Dirichlet series expansions of p-adic L-functions

Abstract

AbstractWe study p-adic L-functions $$L_p(s,\chi )$$ L p ( s , χ ) for Dirichlet characters $$\chi $$ χ . We show that $$L_p(s,\chi )$$ L p ( s , χ ) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of $$\chi $$ χ . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $$c=2$$ c = 2 , where we obtain a Dirichlet series expansion that is similar to the complex case.