Transcendental numbers as solutions to arithmetic differential equations

TL;DR

This paper explores how certain transcendental numbers, including periods, can be solutions to arithmetic differential equations, drawing parallels with algebraic differential equations and suggesting a Galois-theoretic approach to their relations.

Contribution

It introduces the concept that transcendental numbers can solve arithmetic differential equations and proposes a Galois group framework to understand their algebraic relations.

Findings

Some transcendental numbers are solutions to arithmetic differential equations

The approach parallels algebraic differential equations and their solutions

Speculates on Galois groups explaining relations among periods

Abstract

Arithmetic differential equations are analogues of algebraic differential equations in which derivative operators acting on functions are replaced by Fermat quotient operators acting on numbers. Now, various remarkable transcendental functions are solutions to algebraic differential equations; in this note we show that, in a similar way, some remarkable transcendental numbers (including certain "periods") are solutions to arithmetic differential equations. Inspired by a recent paper of Manin, we then speculate on the possibility of understanding the algebraic relations among periods via Galois groups of arithmetic differential equations.