Transcendence and algebraic independence of a family of $p$-adic valuation generating functions

TL;DR

This paper proves the transcendence of a specific $p$-adic valuation generating function over algebraic functions and establishes algebraic independence of its values at certain points, extending Mahler's method beyond automatic functions.

Contribution

It introduces a new transcendence and algebraic independence result for a $p$-adic valuation generating function, expanding Mahler's method to unbounded arithmetic functions.

Findings

Proves $T_p(z)$ is transcendental over $ar{Q}(z)$.

Establishes transcendence of $T_p(z)$ at algebraic points inside the unit disk.

Demonstrates algebraic independence for multiplicatively independent algebraic arguments.

Abstract

We show that $T_p(z)=\prod_{j=1}^{\infty}(1-z^{p^{j}})^{-1/p^{j}}$ is transcendental over $\overline{\mathbb{Q}}(z)$, and establish the transcendence of its values at nonzero algebraic points inside the unit disk. Furthermore, we obtain an algebraic independence result for multiplicatively independent algebraic arguments. In summary, this paper extends Mahler's method beyond the classical automatic setting by studying the function $T_p(z)$, whose coefficients are governed by the unbounded arithmetic function $\nu_p(n)$.