Revisiting Gauss's analogue of the prime number theorem for polynomials over a finite field

In 1901, von Koch showed that the Riemann Hypothesis is equivalent to the assertion that ∑ p≤x p prime 1 = ∫ x 2 dt log t + O( √ x log x). We describe an analogue of von Koch’s result for polynomials over a finite prime field Fp: For each natural number n, write n in base p, say n = a0 + a1 p + · · · + ak pk, and associate to n the polynomial a0 + a1T + · · · + akT k ∈ Fp[T ]. We let πp(X) denote the number of irreducible polynomials encoded by integers n < X, and prove a formula for πp(X) valid with an error term analogous to that in von Koch’s theorem. Our result is unconditional, and is grounded in Weil’s Riemann Hypothesis for function fields. We also investigate an asymptotic expansion for πp(X).