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⩽xpprime1=∫2xdtlogt+O(xlogx). We describe an analogue of von Koch's result for polynomials over a finite prime field FpFp: For each natural number n, write n in base p, sayn=a0+a1p+⋯+akpk,n=a0+a1p+⋯+akpk, and associate to n   the polynomial a0+a1T+⋯+akTk∈Fp[T]a0+a1T+⋯+akTk∈Fp[T]. We let πp(X)πp(X) denote the number of irreducible polynomials encoded by integers n