Competing prime asymptotic densities in \(\mathbb F_q[X]\): a discussion
In the paper under review, the author introduces four densities on the set of prime (irreducible) polynomials over \(\mathbb F_q[X]\). For a positive integer \(n\) let \(I_n\) be the number of prime polynomials of degree \(n\) in \(\mathbb F_q[X]\). Given a set \(S\) of polynomials in \(\mathbb F_q[X]\), let \(S_n\) be the number of prime polynomials of degree \(n\) in \(S\). Definition 1.3 says that \(S\) has density \(d_1,~d_2,~d_3,\) and \(d_4\) if the sequence of general term \[ {{S_n}\over {I_n}},\quad {{\sum_{k\leq n} S_k}\over {\sum_{k\leq n} I_k}},\quad {{1}\over {n}}\sum_{k\leq n} {{S_k}\over {I_k}}\quad {\text{and}}\quad {{\sum_{k\leq n} S_k/q^k}\over {\sum_{k\leq n} I_k/q^k}}, \] respectively, has a limit. In case the corresponding limit exists it is denoted by \(d_i(S)\) for \(i=1,\dots,4\). The main result of the paper is summarized as Theorem A of Section 4. For example, \(d_1\) exists if and only if \(d_2\) exists and whenever they exist for some \(S\), then \(d_1(S)=d_2(S)\). The existence of (either one) of \(d_1(S)\) or \(d_2(S)\) implies the existence of \(d_3(S)\), and the existence of \(d_3(S)\) implies the existence of \(d_4(S)\), which in turn is equivalent to the Dirichlet density. The paper is devoted to the proof of these implications and abounds with examples and an enlightening discussions as to the advantages and disadvantages offered by working with each of the five densities (the four defined above and the Dirichlet density).
relative densities of primes in \(F_q[X]\)
scientific article; zbMATH DE number 5519678