On the p-adic transcendence of ∑_k=1^∞ p^-1/pk

2025/09/29 by Wang, Shanwen, Yuan, Yijun
#11D88 #11J61 #11J81 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.24609

Abstract

Let p be a prime number. In this article, we prove that the p-adic Hahn series ∑_k=1^∞ p^-1/pk, which is the mixed-characteristic analogue of Abhyankar's solution ∑_k=1^∞ t^-1/pk to the Artin-Schreier equation X^p-X-t^-1=0 over F_p( (t) ), is a p-adic complex number, but not a p-adic algebraic number. Based on this result, we formulate a conjecture about the possible order type of the support of an algebraic p-adic Hahn series and prove that it is implied by a tentative observation of Kedlaya.

Citations

• Hyper-algebraic invariants of p-adic algebraic numbers
• On the algebraicity of generalized power series
• Power series and p-adic algebraic closures
• The Algebraic Closure of the Power Series Field in Positive Characteristic

Related

• Some p-adic differential equations
• On p-adic expansions of Ramanujan-like series
• On Summation of p-Adic Series
• Conjectures, consequences, and numerical experiments for p-adic Artin\n L-functions
• Transcendence and algebraic independence of a family of p-adic valuation generating functions