Generating Functions for Extremal Prime Factors in Integers and Exact Moments of the Minimal Factor Degree in F q [ T ] F q [T]

We give explicit Dirichlet series identities for the smallest prime factor pmin⁡pmin and the largest prime factor Pmax⁡Pmax over the integers. We then turn to the polynomial ring Fq[T]Fq[T] and study the degree of the smallest prime factor, dmin⁡dmin. Using the exact generating function of monic polynomials, we obtain closed-form formulas for the survival probability P(dmin⁡>m)P(dmin>m) and for all higher moments of dmin⁡dmin under the uniform distribution on monic polynomials of fixed degree. The formulas are illustrated by a complete worked example over F2[T]F2[T]. All results are self-contained and rigorously proved; no analytic continuation or order-based ambiguities appear.