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The convergence of Euler products over p-adic number fields

Part of: Zeta and $L$-functions: analytic theory Algebraic number theory: global fields

Published online by Cambridge University Press:  23 September 2009

Daniel Delbourgo
Daniel Delbourgo
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, Melbourne 3800, Australia; Email: (daniel.delbourgo@sci.monash.edu.au)
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Abstract

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We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

Keywords

number theoryzeta functionsIwasawa theoryp-adic L-functionsEuler products

MSC classification

Secondary: 11R23: Iwasawa theory 11M41: Other Dirichlet series and zeta functions 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 52 , Issue 3 , October 2009 , pp. 583 - 606
Copyright
Copyright © Edinburgh Mathematical Society 2009