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

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 Xᵖ-X-t⁻¹=0 over Fₚ(\!(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.