Year: 2009
Type: preprint
Abstract: We introduce $p$-adic Kummer spaces of continuous functions on $\mathbb{Z}_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and certain decompositions. As a result, these functions have always a fixed point, functions of certain subclasses have always a unique simple zero in $\mathbb{Z}_p$. The fixed points and the zeros are effectively computable by given algorithms. This theory can be transferred to values of Dirichlet $L$-functions at negative integer arguments. That leads to a conjecture about their structure supported by several computations. In particular we give an application to the classical Bernoulli and Euler numbers. Finally, we present a link to $p$-adic functions that are related to Fermat quotients.
Source: arXiv (Cornell University)
Author: Kellner, Bernd C.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1494625908