アルティン・ハッセの指数関数
A second proof comes from the infinite product for Ep(x): each exponent -μ(n)/n for n not divisible by p is a p-integral, and when a rational number a is p-integral all coefficients in the binomial expansion of (1 - x) are p-integral by p-adic continuity of the binomial coefficient polynomials t(t-1)...(t-k+1)/k! in t together with their obvious integrality when t is a nonnegative integer (a is a p-adic limit of nonnegative integers) . Thus each factor in the product of Ep(x) has p-integral coefficients, so Ep(x) itself has p-integral coefficients.