A generalization of arithmetic derivative to p-adic fields and number fields

Brad Emmons and Xiao Xiao
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 357–382
DOI: 10.7546/nntdm.2024.30.2.357-382
Full paper (PDF, 315 Kb)

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Authors and affiliations

Brad Emmons
Department of Mathematics, Utica University
1600 Burrstone Road, Utica NY 13502, USA

Xiao Xiao
Department of Mathematics, Utica University
1600 Burrstone Road, Utica NY 13502, USA

Abstract

The arithmetic derivative is a function from the natural numbers to itself that sends all prime numbers to and satisfies the Leibniz rule. The arithmetic partial derivative with respect to a prime is the -th component of the arithmetic derivative. In this paper, we generalize the arithmetic partial derivative to -adic fields (the local case) and the arithmetic derivative to number fields (the global case). We study the dynamical system of the -adic valuation of the iterations of the arithmetic partial derivatives. We also prove that for every integer , there are infinitely many elements with exactly anti-partial derivatives. In the end, we study the -adic continuity of arithmetic derivatives.

Keywords

• Arithmetic derivative
• Arithmetic partial derivative
• Arithmetic subderivative
• p-adic fields
• Number fields
• p-adic continuity

2020 Mathematics Subject Classification

• 11A25
• 11R04

References

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Manuscript history

• Received: 20 October 2023
• Accepted: 10 May 2024
• Online First: 27 May 2024

Copyright information

Ⓒ 2024 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).