From automata theory to number theory: p-regularity of p-adic valuations of number theoretic sequences

Abstract

Let p be a prime, νp(x) the p-adic valuation of x, and {νp(f(n))}n≥0 a sequence in Qp generated by the function f : Zp → Qp. This paper examines the convergence properties of functions f(z) in Qp. We study the specific examples of linear recurrent sequences, generated by analytic p-adic functions, and the special factorial sequence of S ⊂ Z, {n!S}n≥0 defined by Bhargava. Here, S ⊂ Z is a union of congruence classes modulo p l. Instead of using standard analytic and algebraic arguments, we use the notion of p-regularity of {νp(f(n))}n≥0 to study the convergence properties of f(n) in Qp. To do so, we determine conditions on f(z) such that {νp(f(n))}n≥0 is p-regular.