Paper on Dirichlet Series Expansions of p-adic L-Functions

Joint paper with Lawrence C. Washington

We study $p$-adic $L$-functions $L_p(s,\chi)$ for Dirichlet characters $\chi$.

We show that $L_p(s,\chi)$ has a Dirichlet series expansion for each regularization parameter $c$ that is prime to $p$ and the conductor of $\chi$. 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 Dirchlet 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 $c=2$, where we obtain a Dirichlet series expansion that is similar to the complex case.

The paper is published in the journal Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. The paper is available here.

Heiko Knospe
Professor für Mathematik in der Nachrichtentechnik

My research interests include number theory, cryptography and network security.