Dirichlet's theorem for the ring of polynomials over \(\mathrm{GF}(2)\)
Let \(G\) denote the ring \(\mathrm{GF}(2)[x]\) of polynomials \(p\) over the field of integers mod 2. Let \(I(k)=\#\{p: \deg p=k\) and \(p\) is irreducible in \(G\}\). A well known analog of the prime number theorem is that \(I(k)=(1/k) \sum_{d\mid k}\mu(d) 2^{k/d}\approx (1/k) 2^k.\) This is extended to an analog of Dirichlet's theorem on primes in arithmetic progressions, with special emphasis on the square-free case. The approach is by way of ``\(L\)-functions'', but in this simpler setting series expansions and coefficient estimates take the place of contour integration.
scientific article; zbMATH DE number 3931145