The distribution of irreducibles in 𝐺𝐹[𝑞,𝑥]
David R. Hayes · Transactions of the American Mathematical Society · 1965
proved for the ring of polynomials over a finite field the following analog of the prime number theorem for arithmetic progressions : Theorem 1.1 (Artin).Let H be a polynomial over a finite field of q elements, and let A be a polynomial prime to H. Ifn(r;H, A) denotes the number of primary irreducibles of degree r which are congruent to A modulo H, then «**o-to"Í+0(£)for some v < 1.A primary polynomial is one whose first coefficient is 1, and <£(//) is the number of polynomials in a reduced residue system modulo H.Let M denote the multiplicative semigroup consisting of the primary polynomials in the ring GF[c7,x] of polynomials over the finite field of q elements, q being a prime power.An equivalence relation on M is said to be a congruence relation if it is compatible with the semigroup structure of M. If 2%H denotes the restriction to M of the relation "congruence modulo H" on GF[q,x], then it is clear that MH is a congruence relation on M for every H in GF[c7,x].Our aim in this paper is to establish a result similar to Theorem 1.1 for a wider class of congruence relations on M than those of the special form â$H. To this end, we have extracted the relevant properties of the relations ¿%H and used these properties to define a class of congruence relations on M which we call the arithmetically distributed relations.The precise definition is given in §8.Theorem 8.1 ofthat section states a result for arithmetically distributed relations which is analogous to that stated in Theorem 1.1 for the relations MH.It includes Theorem 1.1 as a special case.The proof given for Theorem 8.1 is similar to that given by Artin for Theorem 1.1 in that certain analytic functions, the L-functions, are introduced and in that the crucial step of the proof lies in showing that these L-functions do not vanish on the line Real(z) = 1.However, the proof differs from that of Artin in Presented to the Society, August 29, 1963, under the title The distribution of irreducibles in the ring of polynomials over a finite field; received by the editors September 3, 1963.